Part IISigma Freud and

Descriptive Statistics

Chapter 3

Vive La Différence:Understanding Variability

Why Variability Is Important

• How different scores are from one particular score (usually the mean)?– Spread– Dispersion

Measures of Variability

• Three types of variability – Range – Standard Deviation– Variance

• Typically reported together– Average– Variability

Computing the Range

• Most general estimate of variability

• Two types:– Exclusive Range

• r = h - l

– Inclusive Range• r = h – l + 1

Computing the Range

• Data: 98, 86, 77, 56, 48

h 98 highest score - l - 48 lowest score

r 50 range

Types of Range-Like Things

• Range• Mid-range – midpoint of range• Interquartile range – range between 1st & 3rd

quartiles• Semi-interquartile range – midpoint of

interquartile range • Studentized range – range expressed in

standard deviations

Computing Standard Deviation

• Standard deviation (s or SD) - most frequently reported measure of variability

• s or SD = average amount of variability in a set of scores

Computing Standard Deviationby Hand

1. List each score

8

887665543

Computing Standard Deviationby Hand

2. Compute the Mean

8

887665543

60X 10n

606

10

XX

n

Computing Standard Deviationby Hand

3. Subtract the mean from each score

8 8 – 6 = +2

8 8 – 6 = +28 8 – 6 = +27 7 – 6 = +16 6 – 6 = 06 6 – 6 = 05 5 – 6 = -15 5 – 6 = -14 4 – 6 = -23 6 – 3 = -3

Sum 0

Computing Standard Deviationby Hand

4./5. Square each individual difference and sum

8 8 – 6 = +2 +2 x +2 = +4

8 8 – 6 = +2 +2 x +2 = +48 8 – 6 = +2 +2 x +2 = +47 7 – 6 = +1 +1 x +1 = +16 6 – 6 = 0 0 x 0 = 06 6 – 6 = 0 0 x 0 = 05 5 – 6 = -1 -1 x -1 = +15 5 – 6 = -1 -1 x -1 = +14 4 – 6 = -2 -2 x -2 = +43 6 – 3 = -3 +3 x +3 = +9

Sum 0 28

Computing Standard Deviationby Hand

6. Divide the sum by n – 1

7. Compute the square root of 3.11

2( ) 28 283.11

1 10 1 9

X X

n

3.11 1.76SD

Using Excel’s STDEV.S Function

The Computation of the Standard Deviation Using the STDEV.S Function

Using Excel’s STDEV Function

Why n – 1?

• The standard deviation is intended to be an estimate of the POPULATION standard deviation– We want it to be an unbiased estimate– Subtracting 1 from n artificially increases the SD

• A conservative estimate of the population

Comparing the STDEV.S and STDEV.P Functions

Things to Remember…

• Standard deviation is the average distance from the mean

• The larger the standard deviation, the greater the variability

• Standard deviation is sensitive to extreme scores (it is just an Average after all)

Computing Variance

• Variance = standard deviation squared

Using Excel’s VAR.S Function

Comparing the VAR.S and VAR.P Functions

Standard Deviation or Variance

• Standard deviation is stated in original units• Variance is stated in units that are squared

• Which is easier to interpret???

Measures of variability help understand what a distribution of data points looks like.

Use to distinguish distributions from one another and describe a collection of data points.

Summary