Taking the fear out of maths: A guide for psychology teachers · 2021. 1. 12. · taking the fear out of maths: A guide for psychology teachers 6 Int ROD u C t IO n Preparing to teach

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Taking the fear out of maths: A guide for psychology teachers3

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sContentsIntroduction 4

Preparing to teach mathematical skills:

General guidance 6

Mathematical skills 7

Part 1: some basic concepts 9

Rounding to significant figures 9

Percentages 10

Ratios 11

Part 2: types of data 13

Quantitative and qualitative data 13

Primary and secondary data 13

Levels of measurement 14

Part 3: Descriptive 16

Descriptive statistics 16

Measures of central tendency 16

Mean 17

Median 17

Measures of dispersion 18

Range 19

Standard deviation – Interpreting 19

Standard deviation – Formula 20

Standard deviation – How to calculate 21

Standard deviation – Worked example 21

Part 4: Visual displays of quantitative data 23

Visual displays of quantitative data 23

Bar charts 24

Pie charts 26

Line graph 27

Scatter diagram 28

Scatter diagrams –

Understanding correlations 29

Data distributions – Normal distribution 30

Data distributions – Skewed distributions 31

Part 5: Inferential statistics 33

Significance and probability 33

Choosing and using an Inferential

Statistics Test 33

Mann Whitney U Test 34

Wilcoxon Matched Paired

Signed Ranks Test 36

Spearman’s Rank Order

Correlation Coefficient 37

Chi Squared (x2) Test 39

Sign Test 41

Related T Test 42

Unrelated T Test 44

Type 1 and Type 2 errors 45

References 47


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William Thurston Mathematician

IntroductionPsychology and mathematics are two disciplines that are intimately linked. Without mathematics, psychology is toothless; deprived of empirical data and statistical analysis, even the most meticulously constructed psychological theory will never escape the realm of conjecture.

It is the systematic collection and analysis of data that allows psychologists to stand shoulder to shoulder with our colleagues in the traditional sciences of physics, biology and chemistry.

Psychology needs mathematics; however a passion for the former rarely develops into a passion for the latter. Even many psychology teachers have a sense of dread as their

carefully curated schemes of work inch them ever closer to the lessons allocated to statistics and data analysis. But why is there such a fear around the teaching and learning of mathematical skills in psychology? And what can educators do to help both themselves and their students embrace mathematics as the integral part of psychology that it deserves to be?

MAtHEMAtICs Is nOt ABOut nuMBERs, EQuAtIOns, COMPutAtIOn, OR ALGORItHMs: It Is ABOut unDERstAnDInG.


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IOnTaking a student’s eye view, many students who

are wary of mathematics may have struggled with the subject at GCSE and therefore may be turned off from the so-called ‘harder’ sciences such as physics. They may be tempted towards psychology under the impression that the subject will not require mathematical skills. Then, midway through the course, they are introduced to inferential statistics, and the fear descends. Students can also struggle to see the relevance of statistics to both their course and everyday life. Gordon (2004) found that only 7 per cent of psychology students thought that statistics were generally useful in life, and only 16 per cent could see the relevance to their degree.

This fear of mathematics is not just limited to students; many teachers lack confidence in their own knowledge of mathematics, and their perceived ability to deliver mathematical course content to their students. There are some possible reasons for this. Firstly, a significant minority of psychology teachers are not subject specialists, having gained undergraduate degrees in other subjects before teaching GCSE or A-Level psychology. Secondly, even for teachers who are psychology specialists, it may have been many years since their university seminars on data analysis and statistics. Unfortunately, mathematical skills are just not as ‘sticky’ in the memory as much of the other content learned on a psychology degree. Most can probably remember the key features of the Zimbardo Prison Experiment, but the memory of when to use a parametric or non-parametric test has probably long eroded away through neglect.

This lack of confidence with the mathematical component of psychology A-Level can have long term effects on a student’s confidence. A 2014 report commissioned by the Higher Education Academy investigated the attitudes of psychology students towards the mathematical components of their undergraduate degrees. They found that many incoming students felt that they lacked the necessary maths skills from their A-Levels, and anxiety and lack of confidence leads many students to struggle in the statistics components of their course. As all BPS accredited degree programmes must contain a heavy maths component, embedding basic skills in statistics and data handling in post-16 psychology courses will provide huge benefits for those students who go on to study psychology at a higher level. And of course, skills in mathematics are useful for all people, even if a degree in psychology is not in your future!

So how can teachers ensure that they are equipping their students with a sound understanding of statistics, and how can they boost their own confidence in their ability to deliver the mathematical component of A-Level psychology?


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IOn Preparing to teach mathematical skills:

General guidanceThe British Psychological Society (BPS) defines psychology as ‘the science of mind and behaviour’ (BPS, 2014).

The subject is therefore classed as a science subject, and as such, Ofqual have issued regulations for the mathematical content that needs to be included as part of an accredited A-Level in psychology (the full detailed report can be found on Department for Education website here). For all awarding bodies 10% of the marks in psychology must come from mathematics. Each awarding body has interpreted these regulations in slightly different ways, so your first step should be to check your specification carefully. It is likely that your specification will have a section in the appendices where all the required mathematical skills are listed. Make sure that you refer to the specification itself instead of relying solely on a textbook, as it may not include all the skills required.

If using exercises from a textbook or other resource, always make sure that you attempt any mathematical task yourself in advance of the lesson. Do a dry run, ensuring that you are confidently able to follow all the necessary steps and reach the correct answer. This will help you to appear more confident in front of your students, which in turn will help them gain confidence in their own skills.

While it is tempting to be overly positive when trying to sell students on psychology at open evenings, honesty is probably the best policy. Don’t try to hide the maths components of the A-Level. Be honest with students, and allow them to make a fully informed decision over whether psychology A-Level is suitable for them.

When teaching a maths skill for the first time, it is important to assess any prior knowledge that the students may have. Many of the skills required in the A-Level are those that would have been covered in GCSE Mathematics. By gauging the current level of your students, you can tailor your maths lessons to their skill. For example, you may find that you will need to spend a little bit of time brushing up on their understanding of percentages, but bar charts will only need a cursory overview.

It is tempting to teach maths as a single standalone topic. However, students need to understand that maths and stats is an integral part of psychology, not just a side subject that runs parallel to it. Statistics are the bread and butter of psychology, they help us understand what is significant and what is due to chance. Therefore, a way to embed maths skills throughout the course is to continually link to the studies that you learn about. When you look at the results of a study, get students to calculate measures of central tendency themselves. Get them to draw visual representations. Not only will this help cement their understanding, it will also help deepen their memory of the studies that they need to know about as well as helping them understand that statistics are an integral part of psychology.

https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/593849/Science_AS_and_level_formatted.pdf


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The following five sections aim to give you an ‘at a glance’ overview of the core maths skills that both students and teachers often struggle with at A-Level. These are:

Part 1: Some basic concepts

Part 2: Types of data

Part 3: Descriptive statistics

Part 4: Visual displays of quantitative data

Part 5: Inferential statistics

This is not an exhaustive list of every maths skill demanded by the Ofqual awarding body, but it reflects those which turn up the most often throughout the A-Level. For a full list, refer to the Department of Education report, or to the specification of the exam board you are teaching.

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sPart 1: Some basic concepts R O u n D I n G t O s I G n I F I C A n t F I G u R E s

The method of rounding to significant figures is used to simplify numbers to make them easier to understand and compare.

This method can be applied to any kind of number, regardless of size. The term ‘significant figure’ refers to any non-zero number. Examples:

A person’s salary is £28,845. Expressed as two significant figures, this is £29,000

A participant’s reaction time is measured as 0.23487 seconds. Expressed as one significant figure, this is 0.2 seconds

The width of a synapse in the brain is 0.0000251385mm. Expressed as two significant figures this is 0.0000025mm

t O R O u n D t O A s I G n I F I C A n t F I G u R E

• Start at the first non-zero number, and count from there to however many significant figures you require (look at the first digit if counting to one significant figure, include the second if counting to two significant figures and so on)

• Look at the digit after the last significant figure you are rounding to

• If this number is four or less, do not change the previous digit

• If this number is five or more, add one to the previous digit

• Replace the rest of the numbers with zeros. You do not need to replace numbers after a decimal point (these can just be removed)

W O R K E D E X A M P L E

5.66987 to significant figures:

5.66987 – Identify our significant figures (bold)

5.6698 – Look at the next digit. It is more than 5

5.7 – Therefore we add one to the previous digit. The rest of the numbers are removed

When numbers are between integers (whole numbers) we end up with numbers that include decimal places. To simplify what can often be very long numbers, we only include a certain number of decimal places after the decimal point. For example:

66.666666% of students passes a test. Expressed as two decimal places is 66.67%

A person’s reaction time was recorded as 1.344 seconds. Expressed as one decimal place is 1.3 seconds


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s t O R O u n D A D E C I M A L P L A C E

• Start at the decimal point and count to however many digits you require (if counting to one decimal place, only count the first number after the decimal point, if counting to two decimal places, include the second and so on)

• Look at the digit after the last decimal place you are rounding to

• If this number is four or less, do not change the previous digit

• If this number is five or more, add one to the previous digit

• Remove the rest of the numbers after the decimal place you need


2.2332954 to three decimal places:

2.2332954 – Identify our three decimal places (bold)

2.2332954 – Look at the next digit. It is 4 or less

2.333 – Therefore we do not add one to the previous digit. The rest of the numbers are removed

P E R C E n t A G E s

Percentages allow us to easily compare sets of data by expressing a number or ration as a fraction of one hundred. ‘Per cent’ literally means ‘per hundred’. For example:

25% of people who sat an exam gained a B grade (for every 100 people who took the text, 25 of them gained a B grade)

58% of cat owners are female (for every 100 cat owners, 58 of them are female)

C A L C u L A t I n G P E R C E n t A G E s

To calculate percentage, you divide the part you wish to calculate percentage for by the total, then multiply by 100.


Of a class of 30 students, 6 were absent on Monday.

6/30 = 0.2

0.2 x 100 = 20

Therefore, 20% of students were absent on Monday

R E V E R s I n G P E R C E n t A G E s

You may be required to work out a quantity from a given percentage. To do this, you first divide your total by 100, then multiply it by the percentage given.


46 per cent of 350 participants responded ‘yes’ to a survey question.

350/100 = 3.5

3.5 x 46 = 161

Therefore, 161 participants answered ‘yes’


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sC A L C u L A t I n G P E R C E n t A G E C H A n G E

Percentages can also be used to show how much a value has increased or decreased by. To calculate this, take the new value, divide it by the old value, multiply it by 100, and then subtract 100 per cent from this number. This will tell you whether there has been a percentage increase or decrease.


A student scored 53 on a maths test before revision, and 83 after.

83/53 = 1.57

1.57 x 100 = 157%

157% - 100% = 57%

Therefore, their score has gone up by 57%


A patient’s reaction time before a brain surgery was 1.67 seconds, and after was 0.78 seconds

0.98/1.67 = 0.59

0.59 x 100 = 59%

47% - 100% = -41%

Therefore, their reaction time has decreased by 41%

R A t I O s

Ratios allow us to compare the size of one number to another number. For example:

A psychology class has a ratio of male to female students of 1:4 (for every male there are 4 female students)

A difficult exam has a pass/fail ratio of 5:7 (for every 5 students who pass, 7 fail)

You may be asked to simplify a ratio from a set of data. This means to express the ratio between two sets of data to the simplest form possible. To do this, you need to divide both numbers by the largest number that they can both be divided by.


The ratio of students taking psychology and those taking chemistry in a school is 128:48

The largest number that each can be divided by is 16

128/16 = 8

48/16 = 3

The ratio is therefore 8:3, for every 8 psychology students there are 3 chemistry students

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tAPart 2: Types of data Q u A n t I t A t I V E A n D Q u A L I t A t I V E D A t A

Quantitative data is any that is in numerical fom.

E.g. a score on an IQ test, a participant’s reaction time in milliseconds

Qualitative data is non-numerical. It is descriptive in nature, describing qualities or characteristics. It is often produced from interviews and observations.

E.g. a recollection of a memory, an unstructured observation of a child playing

E V A L u A t I O n

Quantitative data provides more options for mathematical analysis. It is easier to compare between data sets and its more likely to be objective. However, it can oversimplify complex behavior.

Qualitative data can provide a deeper insight into phenomena, and it better reflects the complexity of human behaviour and emotions. However, its analysis can be difficult and prone to subjectivity.

P R I M A R y A n D s E C O n D A R y D A t A

Primary data is that which is generated by the researcher themselves for the purposes of the experiment.

E.g. a researcher conducts a questionnaire to assess attitudes to smoking

secondary data is data that was not generated by the researcher for the study, but was initally collected for some other purpose.

E.g. GCSE results, existing medical records

E V A L u A t I O n

Primary data: Researcher has complete control over the nature of the data they collect. However, it can sometimes lack realism and ecological validity.

secondary data: Cheap and quick to use existing data. However, the researcher has no control over how the data was initally collected.


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nominal data: Data that is in categories. We can measure the frequency of occurrence of each category. This is the lowest level of measurement.

E.g. favourite colours, type of pet owned, gender (male/female)

Ordinal data: Data that can be ranked, but there is no equal measure bewteen the ranks. This is the single middle level of measurement.

E.g. position in a race, ranking by attractiveness, a scale of agreement/disagreement

Interval / Ratio: Data where real measurements are used. There is an equal interval between the scores, and the data has mathematical meaning. This is the highest level of measurement.

E.g. number of sweets eaten, score in a memory test, height in centimetres

While ratio data has a true zero (e.g. seconds taken to press a button), interval data does not, and it is possible to go into minus numbers (e.g. tempature in Celsius)

t O P t I P

A good way to remember the levels of data from lowest to highest is nOIR.

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IVE Part 3: Descriptive

D E s C R I P t I V E s t A t I s t I C s

Descriptive statistics give psychologists a way to summarise and describe data. However, they do not allow us to draw any conclusions, nor to a make a firm statement over whether to accept or reject our alternative hypothesis; descriptive statistics do not tell us whether our results are significant. For that, we need to use inferential statistics (more on these later).

However, descriptive statistics are useful in allowing us to compare between sets of data and see at a glance any differences or similarities.

In A-Level psychology, students need to be familiar with the following descriptive statistics, and must be able to calculate them, explain when they would be used, and discuss some of the strengths and weaknesses of them.

Measures of central tendency: Mean, mode and median

Measures of dispersion: Range and standard deviation

M E A s u R E s O F C E n t R A L t E n D E n C y

A measure of central tendency gives us information about the most common or average result in a set of data. Depending upon the type of data you have, different methods. For more detailed explanations of each method, refer to the individual flashcards.

M E A n

The mean is the average of all the data

It can only be used with ratio and interval data

M E D I A n

The median is the middle score of a set of data

It can only be used with ordinal, ratio and interval data

M O D E

The mode is the data point that occurs most often

It can be used with all levels of data (nominal, ordinal, interval and ratio)


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IVEM E A n

W H E n t O u s E

The mean is used to calculate the average score of a set of data. It can only be used on interval and ratio data (not ordinal or nominal).

H O W I t I s C A L C u L A t E D

All of the scores in a set of data are added together, and the total is divided by the number of scores. This gives you the mean.

E X A M P L E

• Scores from 6 participants in a memory test: 57, 46, 87, 45, 53, 74

• 57 + 46 + 87 + 45 + 53 + 74 = 362

• 362/6 = 60.33

• The average score is 60.33

s t R E n G t H s

This is the most sensitive measure as it uses every piece of data.

W E A K n E s s E s

The mean can be easily skewed by a small number of extremely high or low scores; in the example above, adding an extra score of ‘5’ would pull the whole mean down to 52.4.

M E D I A n

W H E n t O u s E

The median tells us the middle score in a set of data. It can be used with ordinal, interval and ratio data (not nominal).


The data is placed into numerical order. With an odd number of data points, the median is simply the middle score. For even numbers, the median is halfway between the middle scores.

E X A M P L E

• Time taken to complete puzzle in seconds: 57, 43, 69, 24, 45, 34

• Put in order: 24, 34, 43, 45, 57, 69

• Middle scores: 43 and 45

• (43 + 45) / 2 = 44

• Median is therefore 44

s t R E n G t H s

Unaffected by extreme values, unlike the mean.

W E A K n E s s E s

Less sensitive as it does not use all the data. Can be misleading in small data sets.


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IVE M O D E

W H E n t O u s E

The mode tells you which score is most common in a set of data. The mode can be used with any level of data (nominal, ordinal, interval and ratio).


Count the occurrences of each number/category. The one with the most is the mode.

E X A M P L E :

• Responses to ‘What is your favourite colour?’ red, blue, red, purple, pink, purple, blue, red

• Red = 3, blue = 2, purple = 2, pink = 1

• Mode is therefore red

s t R E n G t H s

Is the only measure that can be used with nominal data. Unaffected by extreme scores.

W E A K n E s s E s

There may be more than one mode in a set of data. Does not take into account the rest of the data.

It is possible for a set of data to have more than one mode, limiting its usefulness.

M E A s u R E s O F D I s P E R s I O n

Whereas measures of central tendency tell us what the most common or average score is in a set of data, measures of dispersion tell us about how spread out the data is.

• If the scores are all close to the mean, we would say that the data set has low dispersion

• If they are spread out and contain scores that are much higher or lower than the mean, then the data has high dispersion

• There are two measures of dispersion that you need to be familiar with

• The range is a quick and easy method that tells you the absolute spread of your data from the highest to the lowest score

• standard deviation is a method that tells us how clustered or dispersed a set of data is by comparing each score with the mean


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IVER A n G E

W H E n t O u s E

The range is a very simple way to tell the spread of your data. It can allow you to compare the spread of scores between two data sets.


Subtract the lowest score from the highest score. If using an inclusive measure of range, add one to this number.

E X A M P L E

• Highest score = 15, lowest score = 4

• 15 – 4 = 11

• 11 + 1 = 12

• Range is therefore 12

s t R E n G t H s

The range is very quick to calculate.

W E A K n E s s E s

It may be skewed by extreme values. It does not use all pieces of data. It tells you nothing about how widely or tightly spread the data is.

s t A n D A R D D E V I A t I O n – I n t E R P R E t I n G

W H E n t O u s E

Standard deviation (SD) is a measure of dispersion that shows how close together or spread out the data is. The formula calculates the average distance between each set of data and the mean.

H O W t O I n t E R P R E t s t A n D A R D D E V I A t I O n

The higher the number, the more dispersed the data is.

E X A M P L E

• IQ scores of two groups

• Group A: SD = 9.2

• Group B: SD = 7.8

• We can conclude that the IQ scores in Group A are more dispersed than in Group B

s t R E n G t H s

A precise measure of dispersion as it includes all data. Tells us more about the characteristics of the data than the range.

W E A K n E s s E s

May hide characteristics of the data. Can only be used with interval and ratio data.


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IVE s t A n D A R D D E V I A t I O n – F O R M u L A

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stAnDARD DEVIAtIOn

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IVEs t A n D A R D D E V I A t I O n – H O W t O C A L C u L A t E

To calculate standard deviation for a set of data, follow these steps.

1.

• Calculate the mean (add up all scores and divide by number of scores)

2.

• Calculate the raw score mean by subtracting the mean from each of your raw scores

3.

• square (multiply by itself) each of the numbers you got from step 2

s t A n D A R D D E V I A t I O n – W O R K E D E X A M P L E

A psychologist tests the reaction time of 5 participants. Their scores in millisecond are:

321, 514, 298, 396, 401

1.

• Calculate the mean

• 321 + 514 + 298 + 396 + 401 = 1930• 1930 / 5 = 386

2.

• Calculate the raw score mean

• 321 – 386 = -65• 514 – 386 = 128• 298 – 386 = -88• 396 – 386 = -10• 401 – 386 = -15

3.

• square each of the numbers

• -652 = 4225• 1282 = 16,384• -882 = 7744• -102 = 100• 152 = 225

4.

• sum (add) all these numbers together

5.

• Divide this by the number of scores minus 1

6.

• Find the square root of this number. This is your standard deviation

4.

• sum all these numbers together

• 4225 + 16,384 + 7744 + 100 + 225 = 28,678

5.

• Divide this by the number of scores minus 1

• 28,678 / (5-1) = 7,169.5

6.

• Find the square root of this number.

• 7,169.5 = 84.67

Therefore the standard deviation of this set of data is 84.67

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AtAPart 4: Visual displays of

quantitative dataV I s u A L D I s P L A y s O F Q u A n t I t A t I V E D A t A

Being able to display data in a visual format is vital for psychologists. Visual displays allow patterns in data to be spotted more easily and can allow readers to gain a quick but detailed insight into data.

Students need to be comfortable with both interpreting and creating various visual displays of data including: bar charts, histograms, pie charts, line graphs and scatter diagrams, and also need to be familiar with different types of data distributions and related skews.

When creating your own visual representation, some general points to remember:

Make sure that your visual representation has a title, and that axes and keys are labelled clearly with names and numbers

Use an appropriate scale for your axes (e.g. if your highest score is 12, don’t draw an y-axis that goes up to 30!)

Be prepared by going to the exam with a pencil, rubber, ruler and protractor


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B A R C H A R t s

W H E n t O u s E

Bar charts are used for nominal data (data in categories). For this reason each bar is separated from the others, and they do not touch. Because the data is not continuous, there is no set order for the categories to be presented on the x-axis (the horizontal axis).

H O W t O R E A D t H E M

The height of each bar shows the frequency of a category, the higher the bar, the more frequent that particular category is. To find out the precise frequency of each category, read across from the top of the bar to find the number on the y-axis (the vertical axis).

H O W t O D R A W t H E M

Choose an appropriate scale for your y-axis (the highest number on your axis should be at most a little more than your highest frequency). Draw each bar to show its frequency ensuring the bars do not touch.

E X A M P L E

In this example, participants are asked what their favourite takeaway food is.

FAVOURITE FOOD NUMBER OF PEOPLE

Indian 15

Chinese 10

Fish & Chips 17

Pizza 9

Thai 6

A B A R C H A R t s H O W I n G P A R t I C I P A n t ’ s F A V O u R I R E t A K E A W A y F O O D

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H I s t O G R A M s

W H E n t O u s E

Histograms are only used to display continuous data (e.g. height, time etc). In a histogram, there is a clear order that the bars along the x-axis should be displayed, and the bars should touch. They can only be used with interval or ratio data.


As with bar charts, the height of each bar in a histogram shows the frequency within that category, and so can be read in the same way (by looking at the y-axis).


The data must first be grouped into the continuous categories (e.g. if the data is age, the categories could be 30–34 years, 35–39 years etc.). Then, the frequency is counted for each category. Draw each bar to show its frequency ensuring the bars are touching.

E X A M P L E

This example shows the height in centimetres of children in a nursery class.

HEIGHT (CM) NUMBER OF CHILDREN

85-89.9 6

90-94.9 8

95-99.9 12

100-104.9 7

105-109.9 4

A H I s t O G R A M t O s H O W H E I G H t O F n u R s E R y C H I L D R E n I n C E n t I M E t R E s

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100-104.9

105-109.9

85-89.9

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P I E C H A R t s

W H E n t O u s E

Pie charts are another way to show frequency data. Each slice of the pie shows the proportion of a particular category of the data. The total of all the sections of the pie add up to 100 per cent.


The larger the slice, the larger the proportion of the whole. It is useful to practice estimating proportions (e.g. half a pie is 50 per cent, a quarter is 25 per cent, a third is 33 per cent etc).


There are 360 degrees in a circle, so to figure out the angle of each section, take the frequency, divide it by the whole, and then multiply that by 360. In the example below, 8 per cent of the marks are A* grades: (8/100) x 360 = 28.8º. Do this for every slice.

E X A M P L E

Here are the mock exam grades from 100 psychology students.

GRADE NUMBER

A* 8%

A 8%

B 15%

C 23%

D 21%

E 16%

U 9%

A P I E C H A R t t O s H O W G R A D E s G A I n E D I n y 1 3 P s y C H O L O G y M O C K E X A M

U A*A

B

E16%

9% 8%8%

15%

23%21%

DC

WEL

LBEI

NG S

CORE

MONTH

0

5

10

15

20

25

30

35

March

May

July

September

November

January


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L I n E G R A P H

W H E n t O u s E

Like a histogram, a line graph is used for continuous data, but rather than touching bars, the data is shows as a series of points connected with a line. Line graphs are often used to track the changes in a set of data over time.


The height of each data point on a line graph shows the value and so can be read by looking at the y-axis.


Choose an appropriate scale for your y-axis. Plot each data point, and then connect all the points with a line.

E X A M P L E

A school tracks the wellbeing of its students by conducting a survey every two months which gives a wellbeing score.

MONTH WELLBEING

January 9

March 13

May 25

July 29

September 20

November 17

A L I n E G R A P H t O s H O W s t u D E n t W E L L B E I n G

U A*A

B

E16%

9% 8%8%

15%

23%21%

DC

WEL

LBEI

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MONTH

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10

15

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30

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July

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s C A t t E R D I A G R A M

W H E n t O u s E

Scatter diagrams are used to show the relationship or correlation between two co-variables. Each data point on the diagram represents a single participant, and shows their score on each of the two variables measured.


The patterns of the data points can tell you about the type and strength of the relationship between the two variables (see next page for more details).


Draw the x and y axes and label with your variables. For each pair of data, draw a cross or dot where the value on the x-axis intersects the value on the y-axis.

E X A M P L E

An experimenter measures the temperature of a room before giving participants a memory test.

A s C A t t E R G R A P H s H O W I n G t H E R E L A t I O n s H I P B E W t E E n t H E t E M P E R A t u R E O F A R O O M A n D t H E s C O R E I n A M E M O R y t E s t

SCOR

E ON

TES

T

T EMPERATURE IN DEGREES CELSIUS

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

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s C A t t E R D I A G R A M s – u n D E R s t A n D I n G C O R R E L A t I O n s

The pattern of the data points on a scatter diagram can tell you about the relationship between the two variables.

The strength and direction of this relationship can also be expressed as a number: the correlation co-efficient.

• This is a number between -1 and +1. The closer that the correlation coefficient is to either -1 or +1, the stronger the relationship between the two variables

• The – or + before the number tells you whether the relationship is positive or negative

• The correlation co-efficient is calculated by using either Spearman’s Rank Correlation Coefficient, or Pearson’s Correlation Coefficient

P O s I t I V E C O R R E L A t I O n

• As one variable increases, the other variable also increases

• This is a correlation coefficient between 0 and +1

• The closer to +1, the stronger the relationship

• E.g. height and shoe size (the taller you are, the larger your feet)

n E G A t I V E C O R R E L A t I O n

• As one variable increases, the other variable decreases

• This is a correlation coefficient between 0 and -1

• The closer to -1, the stronger the relationship

• E.g. units of alcohol drunk and score in a memory test (the more you drink, the less you remember)

n O C O R R E L A t I O n

• An increase or decrease in one variable has no effect on the other variable

• This is a correlation coefficient close to 0

• E.g. intelligence and hand span (the size of your hands has no relationship with how clever you are)

SCOR

E ON

TES

T

T EMPERATURE IN DEGREES CELSIUS

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

0

5

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65

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10

9

87

6

5

43

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1

00 1 2 3 4 5 6 7 8 9 10

10

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87

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43

2

1

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10

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10

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10

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2

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10

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10

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D A t A D I s t R I B u t I O n s – n O R M A L D I s t R I B u t I O n

When we have large data sets, we may start to see a pattern arise in a data distribution. The larger the data set, the more likely these distributions are to occur. These patterns become most apparent when we plot this data on a frequency chart where the data forms a bell curve (see below). This pattern is called a normal distribution.

In a normal distribution, the majority of the scores are clustered around the average (in a normal distribution, the mean, mode and median are all the same). 68.26 per cent of people will be within one standard deviation of the mean.

As we move further away from the mean, we find fewer and fewer scores. 95.44 per cent are within two standard deviations, 99.74 per cent are within three, meaning that only 0.26 per cent of scores lie beyond three standard deviations.

There are many examples of normal distributions:

• Physical characteristics such as height, foot length or hand span

• Social characteristics such as salary

• Psychological characteristics such as intelligence or reaction time

-3 -2 -1 1 2 3

MEANMODE

MEDIAN

68.26%

95.44%

99.74%

NEGATIVE SKEW

MEAN

MODE

MEDIAN

POSIT IVE SKEW

MEAN

MODE

MEDIAN


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D A t A D I s t R I B u t I O n s – s K E W E D D I s t R I B u t I O n s

Not all sets of data will follow a normal distribution. We will only get a normal distribution if the data is symmetrical – where there is an even distribution of high and low scores. However, sometimes data clusters around very high or very low values. This causes an asymmetrical curve, where one side has a high cluster of scores on one side, and a long narrow tail on the other. These kinds of data distributions are called skewed and a distribution can have either a negative skew or a positive skew.

n E G A t I V E s K E W

In a negative skew, the data is biased towards high scores, with only a handful of extreme low scores. This results in a distribution curve with a cluster of high scores on the right, and few low scores creating a long tail on the left. In a negative skew, the mode is higher than the median, and both are higher than the mean. This is because the handful of low scores are skewing the mean downwards (hence- negative skew).

An example of a negative skew could be a maths test that is far too easy; many students score very high, with the distribution being skewed by a handful of students who performed poorly.

P O s I t I V E s K E W

A positive skew is the opposite – the data is biased towards low scores with a handful of extremely high scores. This results in a distribution curve with a cluster of low scores on the left and few high scores, creating a long tail on the right. In a positive skew, the mode is lower than the median, and both are lower than the mean. This is because the few number of high scores are skewing the mean upwards (hence – positive skew).

An example of a positive skew could be yearly salary. The mean salary in the UK is around £29,000, and most people will take home a wage that is either a little higher or a little lower. However, a tiny percentage of people earn salaries in the hundreds of thousands (or even millions), causing a long tail on the distribution.

-3 -2 -1 1 2 3

MEANMODE

MEDIAN

68.26%

95.44%

99.74%

NEGATIVE SKEW

MEAN

MODE

MEDIAN

POSIT IVE SKEW

MEAN

MODE

MEDIAN

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s I G n I F I C A n C E A n D P R O B A B I L I t y

While descriptive statistics allow us to describe our data, they are unable to tell us whether the results we have found are meaningful or whether they are due simply to chance. To know whether our results are significant we need to carry out an inferential statistics test.

All inferential statistics tests work in the same way – the test analyses our data and compares our results to what we would expect to find if the data were distributed randomly. It then assesses the probability that the results we have found are just due to chance. The further

away our data is from what we would expect just through chance, the more likely it is that our results are meaningful – that we have found something significant.

In psychology, we generally accept a significance level of 5 per cent. This means that there is only a 5 per cent or less probability that our results are due to chance. In other words we can be 95 per cent confident that our results are not due to chance. This is written as p < 0.05.

C H O O s I n G A n D u s I n G A n I n F E R E n t I A L s t A t I s t I C s t E s t

Before you can test your data for significance, you need to decide on which test you need to use. To decide on an appropriate test, this is the information that you need:

P A R A M E t R I C O R n O n - P A R A M E t R I C

A parametric test can only be used when the data is normally distributed. A non-parametric test is used when data is not normally distributed.

L E V E L O F D A t A

Identify the level of data that you have (nominal, ordinal, interval and ratio). Some tests can only be used with nominal data while others are only suitable for ordinal, interval and ratio data.

D I F F E R E n C E O R R E L A t I O n s H I P

Identify whether you are looking for a difference between two sets of data, or a relationship between the data.

I n D E P E n D E n t O R R E L A t E D D A t A

If the data is related, it means that all sets of data come from the same participants (typically when using a repeated measures or correlational design). If the data is independent each participant only gives data for one set of data (when using independent measures design).


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test, you will then use that test to analyse your data (for specific details of each test, look below). However, all tests work roughly in the same way:

1. Follow the mathematical steps of the statistics test using your raw data.

2. At the end of the calculation, you are left with a number. This is your observed value. On its own however, the observed value is meaningless.

3. Using the statistical table for the test you have used, look up the critical value. This is the threshold number that your observed value has to reach in order to be significant.

4. Compare your observed value with the critical value on the table. Depending upon the test used, the observed value needs to either be higher or lower than the critical value to be classed as significant.

5. If your observed value has reached or surpassed the threshold of the critical value, then we say that the result is significant. This means that we accept the alternative hypothesis and reject the null hypothesis. If the result is not significant, we accept the null hypothesis and reject the alternative hypothesis.

M A n n W H I t n E y u t E s t

W H E n t O u s E

• When your data is at least ordinal (ordinal, interval or ratio) and non-parametric

• When you are looking for a difference between two sets of data

• When the data is independent (from using independent measures)

H O W t O I n t E R P R E t R E s u L t s

Once the test has been completed, we are left with our observed value called ‘u’. To know whether this number is significant, we need two pieces of information.

1. Did the study have a one tailed (directional) or two tailed (non-directional) hypothesis?

2. How many participants were in each group? We call these values N1 and N2.

We then use this information to look up the critical value on the statistics table.

• If our observed value of U is equal to or lower than the critical value, our result is significant

• If our observed value of U is higher than the critical value, our result is not significant, and is likely to be due to chance

E X A M P L E

A psychologist compares the map reading ability of male and female participants. She does this by giving the participants a test comprising 20 questions that assess their ability to read maps. She will then compare the scores between the two groups. A Mann Whitney U test was an appropriate choice because:

• The score on the map reading task is at least ordinal

• We are looking for a difference between the results of the male and female participants

• The data comes from two different groups (men and women) and so is independent


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be a difference in the map reading score between male and female participants.’ She uses a sample of 7 men and 6 women, and found that the male participants scored higher than female participants. She conducts a Mann Whitney U test and calculates a u value (observed value) of 8. To look up the critical value, she uses the following:

1. The study had a two tailed hypothesis – the hypothesis was non-directional as she predicted a difference between men and women, but did not state whether men or women would score higher.

2. N1 is 6 and N2 is 7 (the number of males and females in each group).

She uses this to look up the critical value on the statistics table.

• She first chooses the correct table (a two tailed test with p < 0.05)

• She looks down from N1 as 6 and across from N2 as 7

• The critical value is therefore 6

For a Mann Whitney test, the observed value has to be equal to or lower than the critical value to be significant.

Because the observed value (U) of 8 is higher than the critical value of 6, she concludes that her results are not significant, meaning that any difference between the male and female participants in their ability to read maps is due to chance. She therefore will reject her alternative hypothesis and accept the null hypothesis.

N2

CRITICAL VALUES FOR A TWO TAILED TEXT (p < 0.05)

N1

2 3 4 5 6 7 8

2 0

3 0 1 1 2

4 0 1 2 3 4

5 0 1 2 3 5 6

6 1 2 3 5 6 8

7 1 3 5 6 8 10


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W H E n t O u s E



• When the data is related (from using repeated measures)

H O W T O I N T E R P R E T R E S U L T S

Once the test has been completed, we are left with our observed value called ‘t’. To know whether this number is significant, we need two pieces of information.


2. How many participants were in the study. We call this number ‘N’.


• If our observed value of T is equal to or lower than the critical value, our result is significant

• If our observed value of T is higher than the critical value, our result is not significant, and is likely to be due to chance

E X A M P L E

A psychologist wants to see if time of day has an effect on memory. A group of 12 participants are given a memory test in the morning, and then again in the evening. A Wilcoxon’s test was an appropriate choice because:

• The score on the memory task is at least ordinal

• We are looking for a difference between the results of morning and afternoon condition

• Both sets of data (memory scores in the morning and afternoon) come from the same group of participants

The researcher’s hypothesis was ‘Participants will score higher in a memory test in the afternoon than in the morning’. After collecting the data, the researcher found that the afternoon scores were higher. He conducts a Wilcoxon’s test and calculates a t value (observed value) of 15. To look up the critical value, he uses the following:

1. The study had a one tailed hypothesis – the hypothesis was directional as he predicted that scores in the afternoon would be higher.

2. N is 12 (the number of participants).

He uses this to look up the critical value on the statistics table.

• He first chooses the correct table (p < 0.05)

• He looks at the column for one tailed tests where N = 12



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s P E A R M A n ’ s R A n K O R D E R C O R R E L A t I O n C O E F F I C I E n t

W H E n t O u s E


• When you are looking for a correlation between two sets of data

• When the data is related (each participant provides data for each co-variable)


Once the test has been completed, we are left with our observed value also called a ‘correlation co-efficient’. To know whether this number is significant, we need two pieces of information.


2. How many participants were in the study. We call this number ‘N’.

N =ONE

TAILED TESTTWO

TAILED TEST

5 0

6 2 0

7 3 2

8 5 3

9 8 5

10 11 8

11 13 10

12 17 13

13 21 17

14 25 21

For a Wilcoxon’s test, the observed value has to be equal to or lower than the critical value to be significant.

Because the observed value (T) of 15 is lower than the critical value of 17, he concludes that his results are significant, meaning that the difference between the morning and afternoon scores is unlikely to be due to chance. He therefore will reject his null hypothesis and accept the alternative hypothesis.


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critical value on the statistics table.

• If our observed value is equal to or higher than the critical value, our result is significant

• If our observed value is lower than the critical value, our result is not significant, and is likely to be due to chance

E X A M P L E

A researcher wants to see if there is a correlation between the number of Facebook friends a person has and their self-esteem (as measured using a self-esteem questionnaire). A Spearman’s Rank test was an appropriate choice because:

• Both the number of Facebook friends, and the score on the self-esteem questionnaire are at least ordinal

• We are looking for a correlation between the number of friends and self esteem

• Both sets of data come from the same group of participants

The researcher’s hypothesis was ‘There will be a positive correlation between number of Facebook friends and the score on the self-esteem questionnaire’ i.e. the more Facebook friends a person has, the higher their self-esteem. He tests 10 participants. He conducts a Spearman’s Rank test and calculates that the correlation coefficient (observed value) is +0.355. (The + or – before the number shows the direction of the correlation). To look up the critical value, he uses the following:

1. The study had a one tailed hypothesis- the hypothesis was directional as he predicted not only that there would be a correlation, but that it would be a positive one.

2. N is 10 (the number of participants).



• He looks at the column for one tailed tests where N = 10

• The critical value is therefore 0.536

For a Spearman’s Rank, the observed value has to be equal to or higher than the critical value to be significant.

Because the observed value of 0.355 is lower than the critical value of 0.564 (we ignore the +/- sign) he concludes that his results are not significant, meaning that any correlation between Facebook friends and self-esteem is due to chance. He therefore will reject his alternative hypothesis and accept the null hypothesis.

N =ONE

TAILED TESTTWO

TAILED TEST

4 1.00

5 0.900 1.00

6 0.829 0.886

7 0.714 0.738

8 0.643 0.738

9 0.600 0.700

10 0.564 0.648

11 0.536 0.618

12 .0503 0.587

13 0.503 0.560

14 0.464 0.538


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ICsC H I s Q u A R E D ( X 2) t E s t

W H E n t O u s E

• When your data is in categories (nominal data) and non-parametric

• When you are looking for a relationship between two sets of data



Once the test has been completed, we are left with our observed value (x2). To know whether this number is significant, we need two pieces of information.


2. What were the degrees of freedom (df)? Degrees of freedom are calculated by multiplying the number of rows minus 1 by the number of columns minus 1.


• If our observed value is equal to or higher than the critical value, our result is significant

• If our observed value is lower than the critical value, our result is not significant, and is likely to be due to chance

E X A M P L E

A researcher wants to see if there is a relationship between a person’s occupation and whether they own a pet. A Chi-Squared test was an appropriate choice because:

• Both sets of data are nominal (in categories) – occupation, and whether or not they own a pet

• We are looking for an relationship between occupation and pet ownership

• The data is independent – each participant is only in one condition (pet owner or not)

OCCUPATION HAS PETS NO PETS

Lawyer 8 6

Doctor 7 11

Teacher 15 5

Builder 4 12

Retail 11 85

ROW

S

2 COLUMNS


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dfONE

TAILED TESTTWO

TAILED TEST

1 2.71 3.84

2 4.60 5.99

3 6.25 7.82

4 7.78 9.49

5 9.24 11.07

The researcher’s hypothesis was ‘There will be a relationship between occupation and pet ownership’. The results are in the first table. After a Chi-Squared test is conducted, the observed value (x2) is 10.557. To look up the critical value, he uses the following:

1. The study had a two tailed hypothesis – the hypothesis only predicted that there would be an relationship between occupation and pet ownership, not what the relationship would be.

2. Degrees of freedom (df) is 4. There are 5 rows and 2 columns: (5 - 1) x (2 - 1) = 4 x 1 = 4.



• He looks at the column for two tailed tests where df = 4


For a Chi-Squared test, the observed value has to be equal to or higher than the critical value to be significant.

Because the observed value (x2) of 10.557 is higher than the critical value of 9.49, he concludes that his results are significant, meaning that there is a relationship between occupation and pet ownership that is not due to chance. He therefore will reject his null hypothesis and accept the alternative hypothesis.


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W H E n t O u s E

• When your data is in categories (nominal data) and non-parametric




In the Sign test, our observed value (s) is the number of times the least frequent value occurs. To know whether this number is significant, we need two pieces of information.


2. How many scores did we have (minus any zero values)? This is N.


• If our observed value is equal to or lower than the critical value, our result is significant

• If our observed value is higher than the critical value, our result is not significant, and is likely to be due to chance

E X A M P L E

A teacher asks her students how confident they feel before and after a psychology revision session. The Sign test is an appropriate test because:

• The data is nominal (categories): The student can be categorised as feeling more or less confident

• We are looking for an difference between before and after the revision session

• The data is related – each participant is asked about their confidence before and after

The teacher’s hypothesis was ‘Students will report feeling more confident after a revision session’. She delivered the revision session to 12 students, and asked them to rate their confidence out of 10. She then compared the scores before and after, assigning a ‘+’ to students who increased in confidence, and a ‘-’ to those that decreased.

PPTCONFIDENCE

BEFORECONFIDENCE

AFTERCHANGE PPT

CONFIDENCE BEFORE

CONFIDENCE AFTER

CHANGE

1 5 7 + 7 3 4 +

2 7 7 0 8 1 3 +

3 2 4 + 9 5 5 0

4 8 6 - 10 8 9 +

5 9 8 - 11 4 8 +

6 4 6 + 12 9 6 -


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or ‘-’ groups occurs least frequently. In this example, there are 7 ‘+’ and 3 ‘-’. Therefore s = 3. To look up the critical value, the teacher uses the following:

1. The study had a one tailed hypothesis - the hypothesis predicted that student confidence levels would increase after the revision session.

2. N = 10 (12 participants minus the two who did not change in confidence scores).

The teacher uses this to look up the critical value on the statistics table.

• She first chooses the correct table (p < 0.05)

• She looks at the column for one tailed tests where N = 10


For the Sign test, the observed value has to be equal to or lower than the critical value to be significant. Because the observed value (s) of 3 is higher than the critical value of 1, she concludes that her results are not significant, meaning that the revision session did not significantly affect confidence levels. She therefore will reject her alternative hypothesis and accept the null hypothesis.

N =ONE

TAILED TESTTWO

TAILED TEST

5 0

6 0 0

7 0 0

8 1 0

9 1 1

10 1 1

R E L A t E D t t E s t

W H E n t O u s E

• When your data is at least ordinal (ordinal, interval or ratio) and parametric (we assume that the participants are drawn from a normally distributed sample)




Once the test has been completed, we are left with our observed value called ‘t’. To know whether this number is significant, we need two pieces of information.


2. The degrees of freedom (df). To calculate this, you minus 1 from the number of participants.


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critical value on the statistics table.

• If our observed value of t is equal to or higher than the critical value, our result is significant

• If our observed value of t is lower than the critical value, our result is not significant, and is likely to be due to chance

E X A M P L E

A psychologist wants to see if patients with depression respond to a new variation of CBT. She assesses each patient with a depression inventory, and then assesses them again after six weeks of the new therapy. A Related T Test was an appropriate choice because:

• The depressed patients are drawn from a normally distributed sample

• The score on the depression inventory is at least ordinal

• We are looking for a difference between the depression scored before and after therapy

• The two sets of data come from the same group of depressed patients.

The researcher’s hypothesis was ‘Patients will score lower on the depression inventory after receiving CBT.’ She uses a sample of 20 patients, and finds that the average depression score is lower after therapy. She conducts a Related T Test and calculates a t value (observed value) of 1.89. To look up the critical value, she uses the following:

1. The study had a one tailed hypothesis – the hypothesis was directional as she predicted that patients would score lower in the depression inventory after therapy.

2. Degrees of freedom (df) is 19 (20 participants, 20 - 1 = 19).

She uses this to look up the critical value on the statistics table.

• She first chooses the correct table (p < 0.05)

• She looks at the column for a one tailed test where df = 19


For a Related T Test, the observed value has to be equal to or higher than the critical value to be significant.

Because the observed value (t) of 1.89 is higher than the critical value of 1.73, she concludes that her results are significant, meaning that there has been a statistically significant reduction in depression symptoms that is unlikely to be due to chance. She therefore will reject her null hypothesis and accept the alternative hypothesis.

dfONE

TAILED TESTTWO

TAILED TEST

12 1.78 2.18

13 1.77 2.16

14 1.76 2.14

15 1.75 2.13

16 1.75 2.12

17 1.74 2.11

18 1.73 2.10

19 1.73 2.09

20 1.72 2.09

21 1.72 2.09


PA

Rt

5:

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REn

tIA

L s

tAt

Ist

ICs u n R E L A t E D t t E s t

W H E n t O u s E

• When your data is at least ordinal (ordinal, interval or ratio) and parametric (we assume that the participants are drawn from a normally distributed sample)




Once the test has been completed, we are left with an observed value called ‘t’. To know whether this number is significant, we need two pieces of information.

1. Did the study have a one tailed (directional) or two tailed (non-directional) hypotheses?

2. The degrees of freedom (df). To calculate this, you minus 1 from the number of participants in each group and add these two numbers together.


• If our observed value of t is equal to or higher than the critical value, our result is significant.

• If our observed value of t is lower than the critical value, our result is not significant, and is likely to be due to chance.

E X A M P L E

A researcher wants to see the effectiveness of a new drug on the treatment of schizophrenia. 12 schizophrenics are given the new drug, and a control group of 12 schizophrenics are given a placebo drug. Both groups have the severity of their symptoms assessed by a psychiatrist who gives them a score out of 100. An Unrelated T Test was an appropriate choice because:

• The schizophrenic patients are drawn from a normally distributed sample

• The score on the schizophrenia assessment is at least ordinal

• We are looking for a difference between the schizophrenia scores between the two groups

• The two sets of data come from two different groups (drug group and placebo group)

The researcher’s hypothesis was ‘Patients given the drug will score lower on the schizophrenia assessment than patients given the placebo.’ He finds that the schizophrenics given the drug score lower on the assessment. He conducts an Unrelated T Test and calculates a t value (observed value) of 1.36. To look up the critical value, he uses the following:

1. The study had a one tailed hypothesis – the hypothesis was directional as he predicted that schizophrenics given the drug would score lower in the schizophrenia assessment than those given the placebo.

2. Degrees of freedom (df) is 22 (12 participants in each group, (12 - 1) + (12 - 1) = 11 + 11 = 22).


PA

Rt

5:

InFE

REn

tIA

L s

tAt

Ist

ICsHe uses this to look up the critical value on the

statistics table.


• He looks at the column for one tailed tests where df = 22


For a Related T Test, the observed value has to be equal to or higher than the critical value to be significant.

Because the observed value (t) of 1.36 is lower than the critical value of 1.717, the results are not significant. Any improvements in the symptoms of schizophrenia is likely to be due to change and is not because of the new drug. He therefore will reject his alternative hypothesis and accept the null hypothesis.

dfONE

TAILED TESTTWO

TAILED TEST

15 1.753 2.131

16 1.746 2.120

17 1.740 2.110

18 1.734 2.101

19 1.729 2.093

20 1.725 3.086

21 1.721 2.080

22 1.717 2.074

23 1.714 2.069

24 1.711 2.064

t y P E 1 A n D t y P E 2 E R R O R s

While the use of inferential statistical tests aims to allow psychologists to see whether their results are significant or just due to chance, sometimes errors can occur – a result may be found to be significant when it is not, or vice versa. These two types of errors are known as type 1 and type 2 errors.

type 1 errors occur when we find the result to be significant when in fact it is just due to chance. For example, a researcher investigating verbal reasoning may find that boys score higher than girls, and so conclude that boys have better verbal reasoning skills. However, if the sample size was small, or the sample happened to contain a number of highly performing boys, this is not a real finding, but is just due to chance.

type 2 errors happen when we state that the result is just due to chance, when in fact there is a significant finding that we overlook. For example, a researcher concludes that caffeine does not affect the reaction time of her participants, when in fact it does have a small, but significant effect.

RE

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nC

Es


REF

EREn

CEsReferences

British Psychological Society. (2014).

The British Psychological Society [Online].

Available: www.bps.org.uk/public/what-is-psychology

Department for Education. (2014). GCE AS and A level subject content for biology, chemistry, physics and psychology. https://assets.publishing.service.gov.uk/

government/uploads/system/uploads/attachment_data/

file/593849/Science_AS_and_level_formatted.pdf

Field, A.P. (2014). Skills in Mathematics and Statistics in Psychology and tackling transition. The Higher

Education Academy. https://s3.eu-west-2.amazonaws.

com/assets.creode.advancehe-document-manager/

documents/hea/private/resources/tt_maths_

psychology_1568037242.pdf

Gordon, S. (2004). Understanding students’ experiences

of statistics in a service course. Statistics Education Research Journal, 3(1), 40–59.

P I C t u R E R E F E R E n C E s

https://pixabay.com/vectors/distribution-normal-

statistics-159626

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between_mean_and_median_under_different_

skewness.png

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