Psych 230 Psychological Measurement and Statistics Pedro Wolf September 2, 2009

Psych 230

Psychological Measurement and Statistics

Pedro Wolf

September 2, 2009

Previously on “let’s learn statistics in five weeks”

• the logic of research– samples, populations, and variables

• descriptive and inferential statistics– statistics and parameters

• understanding experiments– experimental and correlational studies– independent and dependent variables

• characteristics of scores– nominal, ordinal, interval, and ratio scales– continuous and discrete

Which Scale?

Does the variable have an intrinsic value?

Does the variable have equal values between scores?

Does the variable have a real zero point?

Nominal

YES

Ordinal

NO

YESNO

YESNO

Interval Ratio

Continuous

• A continuous scale allows for fractional amounts – it ‘continues’ between the whole-number amount – decimals make sense

• Examples:– Height– Weight– IQ

Discrete

• In a discrete scale, only whole-number amounts can be measured– decimals do not make sense– usually, nominal and ordinal scales are discrete– some interval and ratio variables are also discrete

• number of children in a family

• Special type of discrete variable: dichotomous– only two amounts or categories– pass/fail; living/dead; male/female

Today….

• Why graphical representations of data?• Stem and leaf plots.• Box plots.• Frequency

– what is it– how a frequency distribution is created

• Graphing frequency distributions– bar graphs, histograms, polygons

• Types of distribution– normal, skewed, bimodal

• Relative frequency and the normal curve– percentiles, area under the normal curve

“… look at the data” (Robert Bolles, 1998)

• Raw data is often messy, overwhelming, and un-interpretable.

• Many data sets can have thousands of measurements and hundreds of variables.

• Graphical representations of data can make data interpretable

• Looking at the data can inspire ideas.

What in the world could these data mean?Imagine over 30,000 observations

Time Lat Long930485:23:06.8600001 32.20497 -111.028930497:04:34.77 32.20482 -111.028930497:04:59.7599998 32.20487 -111.028930497:05:46.7600002 32.20485 -111.029930497:06:05.7600002 32.20578 -111.029930497:06:16.7600002 32.20678 -111.029930497:06:28.7599998 32.20698 -111.028930497:09:31.77 32.20687 -110.999930497:09:58.77 32.2055 -110.993930497:10:07.77 32.20555 -110.992930497:10:37.77 32.20687 -110.986930497:11:38.77 32.20672 -110.979

After plotting those data•By plotting the data and superimposing it on map data, suddenly the previousslide’s data can tell a story

•Of course not all data can tell such a story

• People have developed various ways to visualize their data graphically

Stem and Leaf Plots

5 | 4 6 7 9 9 5

6 | 3 4 6 8 8 5 7 | 2 2 5 6 4 8 | 1 4 8 3 9 | 010 | 6 1N = 18

•data - 54, 56, 57, 59, 59, 63, 64, 66, 68, 72 …

•preserves the data in tact. is a way to see the distribution

•numbers on the left of the line are called the stems and represent the leading edge ofeach of the numbers

•numbers on the right of the line are called the leaves and represent the individual numbers

• indicate their value by completing the stem.

Box Plots•Each of the lines in a box plot represents either quartiles or the range of the data.

•In this particular plot the dots represent outliers.

Frequency distributions - why?

• Standard method for graphing data– easy way of visualizing group data

• Introduction to the Normal Distribution– underlies all of the statistical tests we will be studying

this semester– understanding the concepts behind statistical testing will

make life a lot easier later on

Frequency

Frequency - some definitions

• Raw scores are the scores we initially measure in a study

• The number of times a score occurs in a set is the score’s frequency

• A distribution is the general name for any organized set of data

• A frequency distribution organizes the scores based on each score’s frequency

• N is the total number of scores in the data

Understanding Frequency Distributions

• A frequency distribution table shows the number of times each score occurs in a set of data

• The symbol for a score’s frequency is simply f

• N = ∑f

Raw Scores

• The following is a data set of raw scores. We will use these raw scores to construct a frequency distribution table.

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Frequency Distribution Table

Frequency Distribution Table - Example

• Make a frequency distribution table for the following scores:

5, 7, 4, 5, 6, 5, 4



5, 7, 4, 5, 6, 5, 4

Value Frequency7 1



5, 7, 4, 5, 6, 5, 4

Value Frequency7 1 6 1



5, 7, 4, 5, 6, 5, 4

Value Frequency7 1 6 15 3



5, 7, 4, 5, 6, 5, 4

Value Frequency7 1 6 15 3 4 2



5, 7, 4, 5, 6, 5, 4

X f7 1 6 15 3 4 2

Learning more about our data

• What are the values for N and ∑X for the scores below?

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Results via Frequency Distribution Table

What is N?

N = ∑f


What is ∑X?


What is ∑X?

(17 * 1) = 17

(16 * 0) = 0

(15 * 4) = 60

(14 * 6) = 84

(13 * 4) = 52

(12 * 1) = 12

(11 * 1) = 11

(10 * 1) = 10

__________

Total = 246

Graphing Frequency Distributions


• A frequency distribution graph shows the scores on the X axis and their frequency on the Y axis



• Why?– Because it’s not easy to make sense of this:



• Why?– Because it’s not easy to make sense of this:

• On a scale of 0-10, how excited are you about this class: __________

0=absolutely dreading it 10=extremely excited/highlight of my semester

• Data (raw scores)

5 7 2 3 5 5 5 8 7 7 4 5 10 7 5 4 5 5 7 3 6 2 6 3 5 5 7 2 4 6 3 7 5 5 7 3 5 6 5 5 8 6 7 5 3 5 7 2 3 5 4 5 4 8 3 6 5 5 5 1 2 4 7 5 5 4 3 3 7 5 8 6 3 5 10 0 6 6 3 8 5 4 3 2 4 6 3 7 5 5 7 5 7 5 10 7 5 4 5 5 7 6 3 8 1 5 5 6 4 9 8 5 8 5 7 5 10 7 5 4 5 5 7 4 8 4 5 8 5 5 7 5 5 5 2 4 6 3 7 5 2 4 6 3 7 5 8 6 3 5 10 0 6 7 2 8 8 5 5 8 6 3 6 2 6 3 5 5 7 2 5 10 7 5 4 5 5 7 5 7 5 10 7 5 4 5 5 5 7 2 3 3 7 5 8 6 3 5 10 0 6


X f10 4 9 78 357 40

633

5 434 113 112 31 60 4

Excited about course (0=no,10=yes)

1110987654321

Frequency

50

40

30

20

10

0

0 1 2 3 4 5 6 7 8 9 10



• The type of measurement scale (nominal, ordinal, interval, or ratio) determines whether we use:– a bar graph– a histogram– a frequency polygon

Graphs - bar graph

• A frequency bar graph is used for nominal and ordinal data

Graphs - bar graph


Values on the x-axis

Graphs - bar graph


Frequencies on the y-axis

Graphs - bar graph


In a bar graph, bars do not touch

Graphs - histogram

• A histogram is used for a small range of different interval or ratio scores

Graphs - histogram


Values on the x-axis

Graphs - histogram


Frequencies on the y-axis

Graphs - histogram


In a histogram, adjacent bars touch

Graphs - frequency polygon

• A frequency polygon is used for a large range of different scores

Graphs - frequency polygon

• A frequency polygon is used for a large range of different scores

In a freq. polygon, there are many

scores on the x-axis

Constructing a Frequency Distribution

• Step 1: make a frequency table• Step 2: put values along x-axis (bottom of page)• Step 3: put a scale of frequencies along y-axis (left

edge of page)• Step 4 (bar graphs and histograms)

– make a bar for each value

• Step 4 (frequency polygons)– mark a point above each value with a height for the

frequency of that value– connect the points with lines

Graphing - example

• A researcher observes driving behavior on a road, noting the gender of drivers, type of vehicle driven, and the speed at which they are traveling. Which type of graph should be used for each variable?

• Gender?• nominal: bar graph • Vehicle Type?• nominal: bar graph

• Speed?• ratio: frequency polygon

Use and Misuse of Graphs -2

0

100

200

300

400

500

600

2000 2001 2002 2003

Year

Nu

mb

er

of

Felo

nie

s

Use and Misuse of Graphs

• Which graph is correct?

• Neither does a very good job at summarizing the data

• Beware of graphing tricks

0

100

200

300

400

500

600

2000 2001 2002 2003

Year

Nu

mb

er

of

Felo

nie

s

210

215

220

225

230

235

240

2000 2001 2002 2003

Year

Nu

mb

er

of

Felo

nie

s

Types of Distributions

Distributions

• Frequency tables, bar-graphs, histograms and frequency polygons describe frequency distributions

Distributions - Why?

• Describing the shape of this frequency distribution is important for both descriptive and inferential statistics

• The benefit of descriptive statistics is being able to understand a set of data without examining every score

Distributions : The Normal Curve

• It turns out that many, many variables have a distribution that looks the same. This has been called the ‘normal distribution’.

• A bell-shaped curve

• Symmetrical

• Extreme scores have a low frequency

– extreme scores: scores that are relatively far above or far below the middle score

The Ideal Normal Curve


Symmetrical


Most scores in middle range


Few extreme scores


In theory, tails never reach the x-axis

Normal Curve - height

Height (inches)

80.077.575.072.570.067.565.062.560.057.555.052.5

How tall are you (in inches)?Fr

equ

ency

40

30

20

10

0

Normal Curve - hours slept

Hours of Sleep last night

13121110987654321

Frequency

60

50

40

30

20

10

0

0 1 2 3 4 5 6 7 8 9 10 11 12

Normal Curve - GPA

GPA

4.504.254.003.753.503.253.002.752.502.252.001.75

Frequency

50

40

30

20

10

0

Normal Distributions

• While the scores in the population may approximate a normal distribution, it is not necessarily so for a sample of scores

Height (inches)

72.571.570.569.568.567.566.565.564.563.562.561.5

How tall are you (in inches)? (N=10)

Freq

uen

cy

3.0

2.0

1.0

0.0

Skewed Distributions

• A skewed distribution is not symmetrical. It has only one pronounced tail

• A distribution may be either negatively skewed or positively skewed

• Negative or positive depends on whether the tail slopes towards or away from zero

– the side with the longer tail describes the distribution• Tail on negative side : negatively skewed

• Tail on positive side : positively skewed

Negatively Skewed Distributions


Tail on negative side:Negatively skewed


Contains extremelow scores


Does not contain extreme high scores


Can occur due to a “ceiling effect”

Positively Skewed Distributions


Tail on positive side:Positively skewed


Contains extremehigh scores


Does not contain extreme low scores


Can occur due to a “floor effect”


Rank in Family

654321

Frequency

100

80

60

40

20

0

Bimodal Distributions

• a symmetrical distribution containing two distinct humps

Bimodal - birth month

What month were you born?

Month Born

DecNovOctSepAugJulJunMayAprMarFebJan

Freq

uen

cy25

20

15

10

5

0

Distributions - data

• How many alcoholic drinks do you have per week?



Alcoholic drinks per week

24.522.520.518.516.514.512.510.58.56.54.52.5.5

Frequency

100

80

60

40

20

0



• Positively skewed

Alcoholic drinks per week

24.522.520.518.516.514.512.510.58.56.54.52.5.5

Frequency

100

80

60

40

20

0


• How much did you spend on textbooks for this semester?



Spent on Textbooks ($)

900850

800750

700650

600550

500450

400350

300250

200150

10050

Frequency

60

50

40

30

20

10

0



• Normal – one outlier

Spent on Textbooks ($)

900850

800750

700650

600550

500450

400350

300250

200150

10050

Frequency

60

50

40

30

20

10

0

Kurtosis

• meso- Forming chiefly scientific terms with the sense ‘middle, intermediate’

• lepto- Small, fine, thin, delicate• platy- Forming nouns and adjectives, particularly in biology and

anatomy, with the sense ‘broad, flat’

Relative Frequency and the Normal Curve

Relative Frequency

• Another way to organize scores is by relative frequency

• Relative frequency is the proportion of time that a particular score occurs– remember: a proportion is a number between 0 and 1

• Simple frequency: the number of times a score occurs

• Relative frequency: the proportion of times a score occurs

Relative Frequency - Why?

• We are still asking how often certain scores occurred• Sometimes, relative frequency is easier to interpret

than simple frequency

• Example: • 82 people in the class reported drinking no alcohol weekly

– Simple frequency

• 0.42 of the class (42%) reported drinking no alcohol– Relative frequency

Relative Frequency

• The formula for a score’s relative frequency is:

relative frequency =

f

N

Relative Frequency Distribution

Example

• Using the following data set, find the relative frequency of the score 12

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example

• The frequency table for this set of data is:

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example

• The frequency for the score of 12 is 1, and N = 18

• Therefore, the relative frequency of 12 is:

Example

• The frequency for the score of 12 is 1, and N = 18

• Therefore, the relative frequency of 12 is:

06.018

1

N

ffrequencyrelative

Relative Frequencies

• We can also add relative frequencies together. – For example , what proportion of people scored a passing mark in this exam

(>3):Value Frequency Relative Frequency

6 5 5/18 = 0.285 6 6/18 = 0.334 3 3/18 = 0.173 2 2/18 = 0.112 1 1/18 = 0.061 1 1/18 = 0.06

N=18 Total=1.00

Relative Frequencies

• We can also add relative frequencies together. – For example , what proportion of people scored a passing mark in this exam

(>3): 0.28+0.33+0.17=0.78Value Frequency Relative Frequency

6 5 5/18 = 0.285 6 6/18 = 0.334 3 3/18 = 0.173 2 2/18 = 0.112 1 1/18 = 0.061 1 1/18 = 0.06

N=18 Total=1.00


• When the data are normally distributed (as most data are), we can use the normal curve directly to determine relative frequency.

• There is a known proportion of scores above or below any point• For example, exactly 0.50 of the scores lie above the mean


• The proportion of the total area under the normal curve at certain scores corresponds to the relative frequency of those scores.


• Normal distribution showing the area under the curve to the left of selected scores

Percentiles

• A percentile is the percent of all scores in the data that are at or below a score– Example: 98th percentile - 98% of the scores lie below this.

Homework

• Complete exercises 1, 6, and 9 for chapter 3.• Read chapter 4 and 5 for next week.

Documents

Psych 230 Psychological Measurement and Statistics Pedro Wolf September 2, 2009