Descriptive Statistics: Overview

Descriptive Statistics: Overview

Measures of Center

Mode Median Mean

Measures of Symmetry

Skewness

Measures of Spread

Range

Inter-quartile Range Variance

Standard deviation

Measures of Position

Percentile

Deviation Score

Z-score

Central tendency

• Seeks to provide a single value that best represents a distribution

• Typical measures are

– mode

– median

– mean

Mode

• the most frequently occurring score value

• corresponds to the highest point on the frequency distribution

0 1 2 3 4 5

33 34 35 36 37 38 39 40 41 42 43 44 45

Score

For a given sample N=16:

33 35 36 37 38 38 38 39 39 39 39 40 40 41 41 45

The mode = 39

Mode

• The mode is not sensitive to extreme scores.

0 1 2 3 4 5

33 35 37 39 41 43 45 47 49

Score

For a given sample N=16:

33 35 36 37 38 38 38 39 39 39 39 40 40 41 41 50

The mode = 39

Mode

• a distribution may have more than one mode

0 1 2 3 4 5

33 34 35 36 37 38 39 40

Score

For a given sample N=16:

34 34 35 35 35 35 36 37 38 38 39 39 39 39 40 40

The modes = 35 and

39

Mode

• there may be no unique mode, as in the case of a rectangular distribution

0 1 2 3 4 5

33 34 35 36 37 38 39 40

Score

For a given sample N=16:

33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40

No unique mode

Median

• the score value that cuts the distribution in half (the “middle” score)

• 50th percentile

0 1 2 3 4 5

33 34 35 36 37 38 39 40

Score

Frequency

For N = 15

the median is

the eighth

score = 37

Median

0 1 2 3 4 5

33 34 35 36 37 38 39 40

Score

For N = 16

the median is

the average of

the eighth and

ninth scores =

37.5

Mean

• this is what people usually have in mind when they say “average”

• the sum of the scores divided by the number of scores

Changing the value of a single score may not affect the mode or median, but it will affect the mean.

For a population:

n

∑ X

µ =

For a sample:

n

X ₌ ∑ X

Mean

X=7.07 In many cases the mean is the preferred measure of central

tendency, both as a description of the data and as an estimate of the

parameter.

0 2 4 6 8 10 12 14 16 18

3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5

Nightly Hours of Sleep

__

In order for the mean to be meaningful, the

variable of interest must be measures on an

interval scale.

0 1 2 3 4 5

Score

Frequency

X=2.4

__

Mean

The mean is sensitive to extreme scores and is appropriate for

more symmetrical distributions.

0 1 2 3 4 5

33 34 35 36 37 38 39 40

Score

X=36.8 __

0 1 2 3 4 5

33 34 35 36 37 38 39 40

Score

X=36.5 __

0 5 10 15 20 25 30 35 40

0 20 40 60 80 100 120

140 160

180 200

220 240

Income in 1,000s

X=93.2

__

• a symmetrical distribution exhibits no skewness

• in a symmetrical distribution the

Mean

Median

Mode

0 2 4 6 8 10 12 14 16 18

3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5

Nightly Hours of Sleep

Symmetry

• Skewness refers to the asymmetry of the distribution

0 5 10 15 20 25 30 35 40

0 20 40 60 80 100 120

140 160

180 200

220 240

Income in 1,000s

Skewed distributions

• A positively skewed

distribution is asymmetrical and points in the positive direction.

Mode = 70,000$

Median = 88,700$

Mean = 93,600$

mode mean median

•mode < median < mean

• A negatively skewed distribution

Skewed distributions

0 1 2 3 4 5 6 7

0 20 40 60 80 100

Test score

• mode > median > mean

mode mean

median

Measures of central tendency

+ -

Mode

• quick & easy to compute

• useful for nominal data

• poor sampling stability

Median

• not affected by extreme scores • somewhat poor sampling stability

Mean

• sampling stability

• related to variance

• inappropriate for discrete data

• affected by skewed

distributions

Distributions

• Center: mode, median, mean

• Shape: symmetrical, skewed

• Spread

0 2 4 6 8 10 12 14 16

0 10 20 30 40 50 60 70 80 90 100

Scores

Measures of Spread

• the dispersion of scores from the center

• a distribution of scores is highly variable if the scores differ wildly from one another

• Three statistics to measure variability – range

– interquartile range

– variance

Range

• largest score minus the smallest score

• these two

have same range (80)

but spreads look different

• says nothing about how scores vary around the center

• greatly affected by extreme scores (defined by them)

0 2 4 6 8 10 12 14 16

0 10 20 30 40 50 60 70 80 90 100

Scores

Interquartile range

• the distance between the 25th percentile and the 75th percentile

• Q3-Q1 = 70 - 30 = 40

• Q3-Q1 = 52.5 - 47.5 = 5

• effectively ignores the top and bottom quarters, so extreme scores are not influential

• dismisses 50% of the distribution

0 2 4 6 8 10 12 14 16

0 10 20 30 40 50 60 70 80 90 100

Scores

Deviation measures

• Might be better to see how much scores

differ from the center of the distribution -- using distance

• Scores further from the mean have higher deviation scores

Score Deviation

Amy 10 -40

Theo 20 -30

Max 30 -20

Henry 40 -10

Leticia 50 0

Charlotte 60 10

Pedro 70 20

Tricia 80 30

Lulu 90 40

AVERAGE 50

Deviation measures

• To see how ‘deviant’

the distribution is relative to another, we could sum these scores

• But this would leave us with a big fat zero

Score Deviation

Amy 10 -40

Theo 20 -30

Max 30 -20

Henry 40 -10

Leticia 50 0

Charlotte 60 10

Pedro 70 20

Tricia 80 30

Lulu 90 40

SUM 0

Deviation measures

So we use squared deviations from the mean

Score Deviation

Sq.

Deviation

Amy 10 -40 1600

Theo 20 -30 900

Max 30 -20 400

Henry 40 -10 100

Leticia 50 0 0

Charlotte 60 10 100

Pedro 70 20 400

Tricia 80 30 900

Lulu 90 40 1600

SUM 0 6000

This is the sum of squares (SS)

SS= ∑(X-X)

__

Variance

We take the “average” squared deviation from the mean and call it VARIANCE

(to correct for the fact that sample variance tends to underestimate pop variance)

For a population:

N

= SS

σ

For a sample:

1

2

= −

n

s SS

Variance

1. Find the mean.

2. Subtract the mean from every score.

3. Square the deviations.

4. Sum the squared deviations.

5. Divide the SS by N or N-1.

Score Dev’n Sq. Dev.

Amy 10 -40 1600

Theo 20 -30 900

Max 30 -20 400

Henry 40 -10 100

Leticia 50 0 0

Charlotte 60 10 100

Pedro 70 20 400

Tricia 80 30 900

Lulu 90 40 1600

SUM 0 6000 6000/8

=750

The standard deviation is the square root of the variance

The standard deviation measures spread in the original units of measurement, while the variance does so in units squared.

Variance is good for inferential stats.

Standard deviation is nice for descriptive stats.

Standard deviation

1

2

= −

= n

s SS

s

Example

0 2 4 6 8 10 12 14

0 10 20 30 40 50 60 70 80 90 100

Scores

N = 28 X = 50

s

= 555.55 s = 23.57 N = 28 X = 50

s

= 140.74

s = 11.86

Descriptive Statistics: Quick Review

Measures of Center

Mode Median Mean

* *

Measures of Symmetry

Skewness

Measures of Spread

Range

Inter-quartile Range Variance

Standard deviation

* *

* *

Descriptive Statistics: Quick Review

For a population: For a sample:

Mean

Variance σ

= SS

N

s

= SS n − 1 Standard

Deviation σ = σ

s = s

• Treat this little distribution as a sample and calculate:

– Mode, median, mean

– Range, variance, standard deviation

1 2 3 4 5

Exercise

Descriptive Statistics: Overview

Measures of Center

Mode Median Mean

*

Measures of Symmetry

Skewness

Measures of Spread

Range

Inter-quartile Range Variance

Standard deviation

*

*

Measures of Position

Percentile

Deviation Score Z-score

*

*

Measures of Position How to describe a data

point in relation to its

distribution

Quantile

Deviation Score Z-score

Measures of Position

Quantiles

Quartile

Divides ranked scores into four equal parts 25% 25% 25% 25%

(minimum) (median) (maximum)

Quantiles

10% 10% 10% 10% 10% 10% 10% 10% 10% 10%

Divides ranked scores into ten equal parts

Decile

Quantiles

Divides ranked scores into 100 equal parts

Percentile rank of score x

= • 100 number of scores less than x total number of scores

Percentile rank

Deviation Scores

Score Deviation

Amy 10 -40

Theo 20 -30

Max 30 -20

Henry 40 -10

Leticia 50 0

Charlotte 60 10

Pedro 70 20

Tricia 80 30

Lulu 90 40

Average 50

For a population:

For a sample:

deviation = X − X

deviation = X − µ

•What if we want to compare scores from distributions that have different means and standard deviations?

•Example

–Nine students scores on two different tests

–Tests scored on different scales

Nine Students on Two Tests

Test 1 Test 2

Amy 10 1

Theo 20 2

Max 30 3

Henry 40 4

Leticia 50 5

Charlotte 60 6

Pedro 70 7

Tricia 80 8

Lulu 90 9

Average 50 5

Nine Students on Two Tests

Test 1 Test 2

Deviation Score 1

Deviation Score 2

Amy 10 1 -40 -4

Theo 20 2 -30 -3

Max 30 3 -20 -2

Henry 40 4 -10 -1

Leticia 50 5 0 0

Charlotte 60 6 10 1

Pedro 70 7 20 2

Tricia 80 8 30 3

Lulu 90 9 40 4

Average 50 5

Z-Scores

• Z-scores modify a distribution so that it is centered on 0 with a standard deviation of 1

• Subtract the mean from a score, then divide by the standard deviation

For a population: For a sample:

S X z = X − σ

µ

= X −

z

Z-Scores

Test 1 Test 2 Z- Score 1 Z-Score 2

Amy 10 1 -1.5 -1.5

Theo 20 2 -1.2 -1.2

Max 30 3 -.77 -.77

Henry 40 4 -.34 -.34

Leticia 50 5 0 0

Charlotte 60 6 .34 .34

Pedro 70 7 .77 .77

Tricia 80 8 1.2 1.2

Lulu 90 9 1.5 1.5

Average 50 5 0 0

St Dev 25.8 2.58 1 1

A distribution of Z-scores…

Z-Scores

• Always has a mean of zero

• Always has a standard deviation of 1

• Converting to standard or z scores does not change the shape of the distribution: z scores cannot normalize a non-normal

distribution

A Z-score is interpreted as “number of standard

deviations above/below the mean”

Exercise

Test 3 Z-Score

Amy 52

Theo 39

Max -1.5

Henry 1.3

On their third test, the class average was 45 and the

standard deviation was 6. Fill in the rest.

Descriptive Statistics: Quick Review

For a population: For a sample:

Mean

Variance

Z-score

σ

= SS N

s

= SS n − 1

S X z = X −

σ µ

= X − z

Standard

Deviation σ = σ

s = s

If you add or subtract a constant from each value in a distribution, then

• the mean is increased/decreased by that amount

• the standard deviation is unchanged

• the z-scores are unchanged

If you multiply or divide each value in a distribution by a constant, then

• the mean is multiplied/divided by that amount

• the standard deviation is multiplied/divided by that amount

• the z-scores are unchanged

Messing with Units

Example

Score Dev’s Sq dev Z-score

Theo

5 -1 1 -1.5

Max

3 -3 9 -.5

Henry

5 -1 1 .5

Leticia

7 1 1 .5

Charlotte

7 1 1 1.0

Pedro

8 2 4 -1.0

Tricia

4 -2 4 1.5

Lulu

9 3 9 -.5

MEAN

6

1.94

Adding 1

Score Dev’s Sq dev Z-score

Theo

6 -1 1 -1.5

Max

4 -3 9 -.5

Henry

6 -1 1 .5

Leticia

8 1 1 .5

Charlotte

8 1 1 1.0

Pedro

9 2 4 -1.0

Tricia

5 -2 4 1.5

Lulu

10 3 9 -.5

MEAN

7

1.94

Example

Score Dev’s Sq dev Z-score

Theo

5 -1 1 -1.5

Max

3 -3 9 -.5

Henry

5 -1 1 .5

Leticia

7 1 1 .5

Charlotte

7 1 1 1.0

Pedro

8 2 4 -1.0

Tricia

4 -2 4 1.5

Lulu

9 3 9 -.5

MEAN

6

1.94

Multiplying by 10

Score Dev’s Sq dev Z-score

Theo

50 -10 100 -1.5

Max

30 -30 900 -.5

Henry

50 -10 100 .5

Leticia

70 10 100 .5

Charlotte

70 10 100 1.0

Pedro

80 20 400 -1.0

Tricia

40 -20 400 1.5

Lulu

90 30 900 -.5

MEAN

60

19.4

Other Standardized Distributions

The Z distribution is not the only standardized distribution.

You can easily create others (it’s just messing with units,

really).

Score

Theo 5

Max 3

Henry 5

Leticia 7

Charlotte 7

Pedro 8

Tricia 4

Lulu 9

Average 6 St Dev 1.94

Example:

Let’s change these test

scores into ETS type scores (mean 500, stdev 100)

Other Standardized Distributions

Score Z-Score

ETS type score

Theo 3 -1.5 350

Max 5 -.5 450

Henry 7 .5 550

Leticia 7 .5 550

Charlotte 8 1.0 600

Pedro 4 -1.0 400

Tricia 9 1.5 650

Lulu 5 -.5 450

Average 6 0 500

St Dev 1.94 1 100

Here’s How:

Convert to Z scores Multiply by 100 to increase the st dev Add 500 to increase the mean

Other Standardized Distributions

Exercise

Score Percentile

Deviation

Score Z-Score

IQ type score (Mean 100 Stdev 10)

Theo 20

Max 18

Henry 13

Leticia 17

Charlotte 19

Pedro 16

Tricia 11

Lulu 9