MEASURES OF VARIATION OR DISPERSION

THE SPREAD OF A DATA SET

3 MEASURES OF VARIATION

1) Range: (R)

highest value – lowest value

2) Variance: (s2, 2)

the average of the squares of the distance from the actual mean

3) Standard Deviation: (s, )

the average distance from the actual mean of the data set.

Calculating Variance and Standard Deviation of Listed Data

Example 1: The following data represent the high temperatures recorded over the past week. Find the range, variance and standard deviation.

35, 45, 30, 40, 25, 33, 38

Example 1: Answer

Range: R = 45 – 25 R = 20

*Since variance and standard deviation both represent distances from the mean we must first find the mean of the data set.

35 45 30 40 25 33 38

7246

735.1

Calculating the variance

1. Set up your values in a table. Temp (x)

25

30

33

35

38

40

45

Calculating the variance

2. Subtract the mean from each value.

(reminder: variance and st. dev. are a difference from the mean)

Temp (x) x - 25

30

33

35

38

40

45

-10.1

-5.1

-2.1

-.1

2.9

4.9

9.9

Calculating the variance

3. Square each difference from step 2.

(reminder: variance is the squared difference from the mean)

Temp (x)

x - (x - )2

25 -10.1

30 -5.1

33 -2.1

35 -.1

38 2.9

40 4.9

45 9.9

102.01

26.01

4.41

.01

8.41

24.01

98.01

Calculating the variance

4. Find the mean of the squares in step 3. (variance)

(x - )2

102.01

26.01

4.41

.01

8.41

24.01

98.01

2262.87x

2

2 262.8737.6

7

x

n

Calculating the Standard Deviation

• Standard deviation is merely the square root of the variance.

2 37.6

6.13

Calculating Variance of Grouped Data

Ex 2: The following data represent scores on a 75 point final exam. Find the mean, variance and standard deviation of the data set.

Scores (x) Freq.

10 – 20 2

21 – 31 8

32 – 42 15

43 – 53 7

54 – 64 10

65 – 75 3

Ex 2: answer

1. Find the mean.

Scores (x) Freq. xm fxm

10 – 20 2

21 – 31 8

32 – 42 15

43 – 53 7

54 – 64 10

65 – 75 3

15

26

37

48

59

70

30

208

555

336

590

210

mean: 42.9

Ex 2: answer

2. Subtract mean from each midpoint. 15

26

37

48

59

70

mx mx -27.9

-16.9

-5.9

5.1

16.1

27.1

Ex 2: answer

3. Square each difference from step 2.

-27.9

-16.9

-5.9

5.1

16.1

27.1

778.41

285.61

34.81

26.01

259.21

734.41

mx 2

mx

Ex 2: answer

4. Multiply each squared value by the frequency of that class.

778.41

285.61

34.81

26.01

259.21

734.41

1556.82

2284.88

522.15

182.07

2592.10

2203.23

2

mx 2

mf x

Ex 2: answer

5. Sum up last column and then divide by the total frequency (n).

2

2

( ) 9341.25

9341.25207.6

45

mf x

Ex 2: Answer

6. Standard deviation is merely the square root of the variance.

2 207.6

14.4

Samples vs. PopulationsIn most cases, the variability of a sample will be significantly less than that of the corresponding population….why?

Since samples are often used to represent the variability of an entire population, we must be sure to correct for this bias.

Dividing by n-1 gives us an unbiased estimate of the true population standard deviation or variance.

Calculating Variance of Grouped Data (Sample)

• Formula for grouped:

• This does not alter your calculation for the mean! (still divide by ‘n’)

2

2

1

mf x xs

n

Comparing Standard Deviations

• Whenever two samples have the same units of measure, the variance and standard deviation for each can be compared directly. bus

• Ex: A car dealer wanted to compare miles driven on trade-ins:

• Buicks s = 422 miles• Cadillacs s = 350 miles• Variation in mileage was greater for Buicks!

Comparing Standard Deviations

• However if two samples have different units of measure, the variance and standard deviation must be compared using the coefficient of variation.

• The coefficient of variation is the standard deviation divided by the mean. (x 100%)

For Samples:

100%s

CVarx

For Populations:

100%CVar

Comparing Standard Deviations

Ex 1: The mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5225, and the standard deviation is $773. Compare the variability of the two.

Solution: Find the coefficient of variation of each and whichever number is larger, that set is more variable.

• Coefficient of Variation for sales:

• Coefficient of Variation for commissions:

• Commissions are more variable than the sales!

5var 100% 5.7%

87C

773var 100% 14.8%

5225C

Comparing Standard Deviations

Ex 2: The mean for the number of pages of a sample of women’s fitness magazines is 132, with a variance of 23; the mean for the number of advertisements of a sample of women’s fitness magazine is 182, with a variance of 62. Compare the variations.

Solution: Find the coefficient of variation of each and whichever number is larger, that set is more variable.

• Coefficient of Variation for pages:

• Coefficient of Variation for advertisements:

• Advertisements are more variable than the number of pages!

23var 100% 3.6%

132C

62var 100% 4.3%

182C