1

Distribution Summaries

Measures of central tendencyMeanMedianMode

Measures of spreadRangeStandard Deviation Interquartile Range (IQR)

2

Variance Formula

n

XXn

ii

1

2

2

Definitional Formula

3

Variance Formula

n

n

X

X

n

iin

ii

2

1

1

2

2

Computational Formula

4

Note

2

11

2

n

ii

n

ii XX

5

Other central tendencies

Median

Mode

6

Median

The median is the point in the center of the distribution such that half of the cases are larger and half are smaller

34 54 72 73 96 128 200

54 72 73 96 128 200

7

But, where?

When there are an even number of observations, there is no middle observation.There are two middle observationsThe median is between the two - half way between the two

54 72 73 96 128 200 (73+96)/2 = 84.5

8

What happens?

If top value is raised?

54 72 73 96 128 200

54 72 73 96 128 400

Mean? RaisedMedian? No change

9

What happens?

If bottom value is lowered?

54 72 73 96 128 200

0 72 73 96 128 200 Mean? LoweredMedian? No change

10

Mode

The mode of a distribution is the value that has the most observations.

1 2 4 9 10 10 15 18 19 32

1 2 4 9 10 15 18 19 32

Mode: 10

No mode

11

Compare for cohabit length Mos| Freq. 0 | 246 1 | 25 2 | 18 3 | 6 4 | 12 5 | 7 6 | 8 7 | 17 8 | 14 9 | 16 10 | 19 11 | 21 12 | 14 13 | 9 14 | 11 15 | 16 16 | 7 17 | 5 18 | 8

19 | 11 20 | 5 21 | 10 22 | 8 23 | 10 24 | 9 25 | 9 26 | 5 27 | 1 28 | 4 29 | 3 30 | 2 31 | 5 32 | 4 33 | 2 34 | 5 35 | 2 36 | 3 37 | 4 39 | 2

40 | 1 41 | 2 43 | 1 44 | 2 45 | 1 46 | 5 47 | 2 48 | 2 50 | 2 51 | 2 56 | 1 57 | 1 62 | 4 63 | 1 66 | 2 67 | 1 71 | 2 73 | 2 74 | 1

76 | 1 79 | 1 81 | 2 97 | 2 103 | 2 ------------Tot | 626

Mode: 0

Median: 5

Mean: 11.75

12

GraphicallyFr

acti

on

# Months Cohabited0 5 10 15 20 25 30 35 40 45 50 103

0

.461661

ModeMedian

Mean

13

AnotherFr

acti

on

llx.6 .7 .8 .9 1

0

.188

14

Central tendency

Mean: find the “average” value

Median: find the “middle” value

Mode : find the “most common” value

15

Measures of Spread

Range

Variance

Standard deviation

Interquartile range IQR

16

Quartile

Median is the half-way point -- half of cases are above, half are below

Quartile is the quarter point

There are 3 quarter pointsFirst quartile 25% belowSecond quartile 50% belowThird quartile 75% below

17

Consider cohabitation# Months |Cohabited | Freq. Percent Cum.------------+----------------------------------- 0 | 246 39.30 39.30 1 | 25 3.99 43.29 2 | 18 2.88 46.17 3 | 6 0.96 47.12 4 | 12 1.92 49.04 5 | 7 1.12 50.16 6 | 8 1.28 51.44 . . . 13 | 9 1.44 69.01 14 | 11 1.76 70.77 15 | 16 2.56 73.32 16 | 7 1.12 74.44 17 | 5 0.80 75.24 18 | 8 1.28 76.52

18

Graphically

# Months Cohabited0 103

0 5 17

19

Interquartile range (IQR)

Difference between the third quartile and the first quartile, Q3 - Q1

Compare with the range - the difference between the maximum and the minimum, Max - Min

20

Boxplot

Graph based on median and IQR

Lower end of box: Q1

Upper end of box: Q3

Middle line: Q2 (median)

Upper whisker: 1.5*IQR above Q3

Lower whisker: 1.5*IQR below Q1

Outliers: plotted separately

21

Depression Boxplot

ces-d scale items s112-s1230 71

5 10 16.5

IQR = 16.5 - 5 = 11.5

1.5*IQR = 17.25

Upper whisker = 16.5 + 17.25 = 33.75

22

Scatterplot with boxesces-

d s

cale

ite

ms

s112-s

123

Rosenberg self-esteem scale10 40

0

71

Dep

ress

ion

Esteem