Basic Elements of Descritive Statistic

7/28/2019 Basic Elements of Descritive Statistic

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What is Statistics?

Statistics is the science of describing ormaking inferences about the world from asample of data.

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Descriptive Inferential

Statistics

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Descriptive Statistics

Descriptive statistics are methods fororganizing and summarizing data.

For example, tables or graphs are used toorganize data, and descriptive values areused to summarize data.

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Inferential Statistics

Two main methods:1. Estimation

The sample statistic is used to estimate a

population parameter.

A confidence interval about the estimate is

constructed.

2. Hypothesis testing

A null hypothesis is put forward.

Analysis of the data is then used to

determine whether to reject it.4


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Definitions

A variable is a characteristic or conditionthat can change or take on different

values.

Datum is one observation about thevariable being measured.

Data are a collection of observations.

The goal of statistics is to help researchersorganize and interpret the data.

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TYPES OF DATA

VARIABLES

QUANTITATIVEQUALITATIVE

RATIO INTERVALORDINAL NOMINAL

Discrete Continuous

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Nominal or categorical data is data that comprises of

categories that cannotbe rank ordered each category is just

different.

The categories available cannot be placed in any order and nojudgement can be made about the relative size or distance from

one category to another.

What does this mean? No mathematical operations can be

performed on the data relative to each other.

Therefore, nominal data reflect qualitative differences rather

than quantitative ones.

Nominal data

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Examples:

Nominal data

What is your gender?

(please tick)

Male

Female

Did you enjoy the film?

(please tick)

Yes

No

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Systems for measuring nominal data must ensure that

each category is mutually exclusive and the system of

measurement needs to be exhaustive.

Variables that have only two responses i.e. Yes or No,

are known as dichotomies.

Nominal data

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Ordinal data is data that comprises of categories that

can be rank ordered.

Similarly with nominal data the distance between each

category cannot be calculated but the categories can be

ranked above or below each other.

What does this mean? Can make statistical judgements

and perform limited maths.

Ordinal data

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Example:

Ordinal data

How satisfied are you with the level of

service you have received? (please tick)

Very satisfied

Somewhat satisfied

Neutral

Somewhat dissatisfied

Very dissatisfied

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Both interval and ratio data are examples of scale data.

Scale data:

data is in numeric format (50, 100, 150)

data that can be measured on a continuous scale

the distance between each can be observed and

as a result measured the data can be placed in rank order.

Interval and ratio data

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Interval data measured on a continuous scale and has no

true zero point. Examples:

Time moves along a continuous measure or

seconds, minutes and so on and is without a zero

point of time.

Temperature moves along a continuous measure of

degrees and is without a true zero.

Interval data

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Ratio data measured on acontinuous

scale anddoes

have a true zeropoint. Examples:

Age

Weight

Height

Ratio data measured on a discrete scale and does have a true zero point.

Example:

Number of children

Ratio data

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These levels of measurement can be placed in hierarchical order.

Hierarchical data order

Ratio

Interval

Ordinal

Nominal

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Population

The entire group of individuals is called the

population.

Population

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Sample

Usually populations are so large that a

researcher cannot examine the entire group.Therefore, a sample is selected to representthe population in a research study. The goal

is to use the results obtained from thesample to help answer questions about thepopulation.

Population

Sample

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Why sample?

Measuring all units (trees, products, birds, etc.) is

impractical, if not impossible.

Sampling just a few units saves money.

Sampling just a few units saves time.

Some measurements are destructive:

cutting down trees to inspect ring patterns or stem analysiscapturing wildlife to examine their morphology, etc.

Sampling makes statistical methods attractive and powerful.

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A descriptive value for a population iscalled a parameterand a descriptive

value for a sample is called astatistic.

Parameterversus Statistic

PopulationSample

ParameterStatistic

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Statistic tools


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Tables

One way frequency table

Number of passangers Frequency

2 2

4 23

5 41

6 18

7 88 1

For nominal, ordinal and discrete variables.

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Two way frequency table

Tables

Sex\ Hobby Dance Sports TV Total

Men 2 10 8 20

Women 16 6 8 30

Total 18 16 16 50


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Frequency table

Tables

Age Frequency Percentage

10-14 2 5

15-19 16 40

20-24 18 45

25-29 37.5

30-34 1 2.5

Total 40 100

For quantitive variables.

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Graphs

Bar chart

Pie chart

Pictograms

HistogramDensity plot

Scatter plot

Time series plotBoxplot

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Graphs


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Graphs


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Graphs

Statistic pictograms

Do not recommended

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Graphs

Only for numerical variables

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Graphs


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Graphs


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Graphs examples on web

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Recommended book

http://www.laeditorialvirtual.com.ar/Pages2/Huff_Darrell/Huff_ComoMentirConEstadisticas.html#_Toc334380216

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http://www.laeditorialvirtual.com.ar/Pages2/Huff_Darrell/Huff_ComoMentirConEstadisticas.htmlhttp://www.laeditorialvirtual.com.ar/Pages2/Huff_Darrell/Huff_ComoMentirConEstadisticas.htmlhttp://www.laeditorialvirtual.com.ar/Pages2/Huff_Darrell/Huff_ComoMentirConEstadisticas.html


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A cartoon

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Recommended videos

http://www.youtube.com/watch?v=nUJNstRFvvo

http://www.youtube.com/watch?v=ETbc8GIhfHo

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http://www.youtube.com/watch?v=nUJNstRFvvohttp://www.youtube.com/watch?v=ETbc8GIhfHohttp://www.youtube.com/watch?v=ETbc8GIhfHohttp://www.youtube.com/watch?v=ETbc8GIhfHohttp://www.youtube.com/watch?v=nUJNstRFvvohttp://www.youtube.com/watch?v=nUJNstRFvvohttp://www.youtube.com/watch?v=nUJNstRFvvo


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A measure of central tendency is a value that represents a

typical, or central, entry of a data set. The three most

commonly used measures of central tendency are the mean,

the median, and the mode.

Measures of Central Tendency

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The mean of a data set is the sum of the data entries divided

by the number of entries.

Population mean:x

N

Sample mean:

xx

n

mu x-bar

Mean

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Calculate the population mean.

Mean

N

x

7

343

49 years

53 32 61 57 39 44 57

Example: the following are the ages of all seven

employees of a small company:

The mean age of the employees is 49 years.

Add the ages and divide by 7.

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Median

The median of a data set is the value that lies in the middleof the data when the data set is ordered. If the data set has

an odd number of entries, the median is the middle data

entry. If the data set has an even number of entries, the

median is the mean of the two middle data entries.

53 32 61 57 39 44 57To find the median, sort the data.

Example: calculate the median age of the seven employees.

32 39 44 53 57 57 61

The median age of the employees is 53 years.39


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The mode is 57 because it occurs the most times.

Mode

The mode of a data set is the data entry or category thatoccurs with the greatest frequency. If no entry is repeated,

the data set has no mode. If two entries occur with the same

greatest frequency, each entry is a mode and the data set is

called bimodal.

53 32 61 57 39 44 57

Example: find the mode of the ages of the seven employees.

An outlieris a datum that is far from the other in the data set.

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53 32 61 57 39 44 57 29

Recalculate the mean, the median, and the mode. Whichmeasure of central tendency was affected when this new

age was added?

Mean = 46.5

Example: A 29-year-old employee joins the company

and the ages of the employees are now:

Comparing the Mean, Median and Mode

Median = 48.5

Mode = 57

The mean takes every value into

account, but is affected by the outlier.

The median and mode are not

influenced by extreme values.

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Weighted Mean

A weighted mean is the mean of a data set whose entries

have varying weights. A weighted mean is given by

where wis the weight of each entryx.

( )x wx

w

Example: grades in a statistics class are weighted as

follows.

Tests are worth 50% of the grade, homework is worth 30% ofthe grade and the final is worth 20% of the grade. A student

receives a total of 80 points on tests, 100 points on

homework, and 85 points on his final. What is his current

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Weighted Mean

Source Score,x Weight, w xw

Tests 80 0.50 40Homework 100 0.30 30

Final 85 0.20 17

The students current grade is 87%.

( )x wxw

87100

0.87

Begin by organizing the data in a table.


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Shapes of Distributions

A frequency distribution is symmetric when a vertical line can be drawnthrough the middle of a graph of the distribution and the resulting halves

are approximately the mirror images.

A frequency distribution is uniform (orrectangular) when all entries, or

classes, in the distribution have equal frequencies. A uniform distribution

is also symmetric.

A frequency distribution is skewed if the tail of the graph elongates more

to one side than to the other. A distribution is skewed left (negatively

skewed) if its tail extends to the left. A distribution is skewed right

(positivelyskewed) if its tail extends to the right.


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Mean > Median > ModeMean < Median < Mode

Summary of Shapes of Distributions

Mean = Median

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Measures of Variation

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The mean is a good indicator of the central tendency of a setof data, but it does not provide the whole picture about the

data set.

Example 1: comparison of the distribution of two data sets

Mean Median

Data set A: 5 6 7 8 9 7 7Data set B: 1 2 7 12 13 7 7

Note: Both the distributions have same mean and median, butbeyond that they are quite different. In the distribution A, 7 is afairly typical value but in distribution B, most of the values differquite a bit from 7. What is needed here is some measure ofthe dispersion or spread of the data. Following example willillustrate further the importance of measuring the variability in adata set.

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Example 2: Suppose that in a hospital, each patients pulserate is taken in the morning, at noon, and in the evening. On a

certain day, pulse rate for

Mean Median

Patient A: 72 76 74 74 74Patient B: 72 91 59 74 72

Note: Mean pulse rate is same for both the patients. While

patientAs pulse rate is stable, patient Bs fluctuates widely.

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Range

The range of a data set is the difference between themaximum and minimum date entries in the set.

Range = (Maximum data entry) (Minimum data entry)

Example:

The following data are the closing prices for a certain

stock on ten successive Fridays. Find the range.

Stock 56 56 57 58 61 63 63 67 67 67

The range is 67 56 = 11.


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Population Variance and Standard Deviation

The populationvariance of a population data set ofN

entries is

Population variance =

sigma squared

The populationstandard deviation of a population data set

ofNentries is the square root of the population variance.

Population standard deviation =

sigma

2 =( )2

=( )2

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Sample Variance and Standard Deviation

The samplevariance of a sample data set ofn entries is

Sample variance =

S squared

The samplestandard deviation of a sample data set ofn

entries is the square root of the sample variance.

Sample standard deviation =

S

2 =( )2

1

2 =( )2

1

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Interpreting Standard Deviation

When interpreting standard deviation, remember that is a

measure of the typical amount an entry deviates from the

mean. The more the entries are spread out, the greater the

standard deviation.

10

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6

4

2

0

Data value

Frequency

1214

2 4 6

= 4

s = 1.18

x

10

8

6

4

2

0

Data value

Frequency

1214

2 4 6

= 4

s = 0

x

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Measures of Position

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Quartiles


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Quartiles

The three quartiles, Q1, Q2, and Q3, approximately divide anordered data set into four equal parts.

Median

0 5025 10075

Q3Q2Q1

Q1 is the median of the

data below Q2.

Q3 is the median of

the data above Q2.

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Finding Quartiles


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Finding Quartiles

Example:The quiz scores for 15 students is listed below. Find the first,

second and third quartiles of the scores.

28 43 48 51 43 30 55 44 48 33 45 37 37 42 38

Order the data.

28 30 33 37 37 38 42 43 43 44 45 48 48 51 55

Lower half Upper half

Q2Q1 Q3

About one fourth of the students scores 37 or less; about one half

score 43 or less; and about three fourths score 48 or less.

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Interquartile Range


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Interquartile Range

The interquartile range (IQR) of a data set is the differencebetween the third and first quartiles.

Interquartile range (IQR) = Q3Q1.

Example:

The quartiles for 15 quiz scores are listed below. Find the

interquartile range.

(IQR) = Q3Q1

Q2 = 43 Q3 = 48Q1 = 37

= 48 37

= 11

The quiz scores in the middle

portion of the data set vary by at

most 11 points.

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Box and Whisker Plot (boxplot)


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Box and Whisker Plot (boxplot)

A box-and-whisker plot is an exploratory data analysis tool

that highlights the important features of a data set.

The five-number summary is used to draw the graph.

The minimum entry

Q1 Q2 (median)

Q3 The maximum entry

Example:

Use the data from the 15 quiz scores to draw a box-and-

whisker plot.

Continued.

28 30 33 37 37 38 42 43 43 44 45 48 48 51 55

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Box and Whisker Plot


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Box and Whisker Plot

Five-number summary

The minimum entry

Q1 Q2 (median)

Q3 The maximum entry

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28

55

4348

40 44 48 52363228 56

28 37 43 48 55

Quiz Scores

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Parts of a boxplot


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Parts of a boxplot

Percentiles and Deciles


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Percentiles and Deciles

Percentiles divide an ordered data set into 100 parts.There are 99 percentiles: P1, P2, P3P99.

Deciles divide an ordered data set into 10 parts. Thereare 9 deciles: D1, D2, D3D9.

Example: A test score at the 80th percentile (D8), indicatesthat the test score is greater than 80% of all other test scores

and less than or equal to 20% of the scores.

Design matrix


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Design matrix

Sex Age Smoke Country Married

Female 23 Yes USA Yes

Male 43 Yes Colombia Yes

Male 19 Not Brazil Yes

Male 23 Yes Brazil Not

Female 56 Not Canada Yes

Female 78 Yes USA Yes

Male 54 Not Spain Not

Male 76 Yes Colombia Not

Female 43 Not Peru Yes

5 Variables

10 Individuals

Dimension 10 x 5

Documents

Basic Elements of Descritive Statistic