Measures of Dispersion

Measures of Dispersion ◦ How spread out are the data? ◦ Describing quantitative data will not be complete without knowing how observed values are spread out from the average. ◦ For example, two classes who sat the same exam might have the same mean mark but the marks may vary in a different pattern around this. ◦ Measures include 1.

2.

3.

Range Variance Standard Deviation (SD) Pharmaceutical Biostatistics: Descriptive Statistics

Measures of Dispersion n  Example: Table 1.3: Individual values associated with two sets of data possessing identical means.

Data Set A

Data Set B

10

28

20

29

30

30

20

29

10

28

Mean=30

Mean=30

Pharmaceutical Biostatistics: Descriptive Statistics

a measure of the extent to which the observed values are spread out from the averag

2.1We The Range will consider several measures of dispersion and discuss the merits and pitfalls of e

Range

One very simple measure of dispersion is the range. Lets consider the two distribut 2.1 The Range given in Figures 3 and 4. They represent the marks of a group of thirty students on ests. ◦ Very simple measure of dispersion.

One very simple measure of dispersion is the range. Lets consider the two distribut Difference the maximum and of the given in ◦ Figures 3 andbetween 4. They represent the marksvalue of a group thirty students on minimum value (max-min). tests. ◦ Example 1) Marks 40 on test A 50

40

70

Figure 3: Marks on test A. 50

2) Marks on test B

40

60

60

70

80

80

Figure 3: Marks on test A.

50

60

70

80

Pharmaceutical Biostatistics: Descriptive Statistics

40

Figure504: Marks B. 60 on test 70

80

simple measure of dispersion is the range. Lets consider the two distributions 1veryThe Range

n in Figures 3 and 4. They represent the marks of a group of thirty students on two s. very simple measure of dispersion is the range. Lets consider the two distributions ne

Calculating Range‌

ven in Figures 3 and 4. They represent the marks of a group of thirty students on two sts.

1)  Marks on test A 40

40

50

60

70

50 3: Marks 60 on test 70 A. Figure

80 80

2)  Marks onFigure test B3: Marks on test A.

40

50

60

70

80

On test A, the range of marks is 70-45=25.

40 50 4: Marks 60 on test 70 B. 80 Figure On test B, the range of marks is 65-45=20.

Figure on test B. out than the marks on test B e it is clear that the marks on test 4:A Marks are more spread we need a measure of dispersion that will accurately indicate this. ere it is clear that the marks on test A are more spread out than the marks on test B, Pharmaceutical Biostatistics: Descriptive Statistics

wever, as a measure of dispersion the range is severely limited. Since it depends only on observations, the lowest and the highest, we will get a misleading idea of dispersion if se values are outliers. This is illustrated very well if the students’ marks are distributed n Figures 5 and 6.

Range (another example) 1) Marks on test A

hematics Learning Centre, University of Sydney 40

50

60

70

80

6

Figure 5: Marks on test A. 2) Marks on test B

40

50

60

70

80

On test A, the range of marks is 70-45=25.

Figure 6: Marks on test B. On test B, the range of marks is 72-40=32.

test A, the range is still 70 − 45 = 25.

Pharmaceutical Biostatistics: Descriptive Statistics

test B, the range is now 72 − 40 = 32, but apart from the outliers, the distribution of

Variance ◦ Squared deviations from the mean. ◦ The sample variance of a set of n sample data is the number (s2) defined by the formula:

◦ The population variance (σ2) formula :

Pharmaceutical Biostatistics: Descriptive Statistics

Standard Deviation ◦ Measure of variation (deviation) of all values from the mean. ◦ Positive square root of the variance. ◦ Properties include: ◦ The value is usually positive. ◦ 0 indicates no variation. ◦ Larger values indicate greater variation. ◦ The value can increase dramatically with the inclusion of one or more outliers. ◦ The units are the same as the units of the original data values.

Pharmaceutical Biostatistics: Descriptive Statistics

Standard Deviation ◦ Population standard deviation:

◦ Sample standard deviation

Pharmaceutical Biostatistics: Descriptive Statistics

Procedure for finding the standard deviation (sample) ◦ Step 1: Compute the mean ◦ Step 2: Subtract the mean from each individual value to get a list of deviations of the form (x – mean) ◦ Step 3: Square each of the differences obtained from step 2 (x – mean)2 ◦ Step 4: Add all of the squares obtained from step 3 [∑ (x – mean)2] ◦ Step 5: Divide the total from step 4 by the number (n – 1), which is 1 less than the total number of values present. ◦ Step 6: Find the square root of the result of step 5.

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◦ E.g.: Multiple waiting line: 1, 3, 14 ◦ Step 1: mean = 18/3 = 6.0 min ◦ Step 2: e.g. 1 - 6 = -5 ◦ Step 3: e.g. (-5)2 = 25 ◦ Step 4: 25 + 9 + 64 = 98 ◦ Step 5: 98/3-1 = 49 ◦ Step 6: √49 = 7.0 min

x

x – mean

(x – mean)2

1 3

-5 -3

25 9

14 Total: 18

8

64 98

Pharmaceutical Biostatistics: Descriptive Statistics

Range Rule of Thumb ◦ For interpreting a known value of the standard deviation ◦ Minimum “usual” value = (mean) – 2 x standard deviation ◦ Maximum “usual” value = (mean) + 2 x standard deviation

Pharmaceutical Biostatistics: Descriptive Statistics

Example ◦ Past results from the National Health Survey suggest that the head circumferences of 2 months old girls have a mean of 40.05 cm and a standard deviation of 1.64 cm. Determine whether a circumference of 42.6 cm would be considered “unusual”.

Pharmaceutical Biostatistics: Descriptive Statistics

Solution ◦ With a mean of 40.05 cm and a standard deviation of 1.64 cm, we use the range rule of thumb to find the minimum and maximum usual circumferences as follows: ◦ Minimum usual value = (mean) – 2 x (standard deviation) = 40.05 – 2 x 1.64 = 36.77 cm ◦ Maximum usual value = (mean) + 2 x (standard deviation) = 40.05 + 2 x 1.64 = 43.33 cm ◦ Based on these results, we expect that typical twomonth-old girls have head circumferences between 36.77 cm and 43.33 cm. Because 42.6 cm falls within those limits, it would be considered usual or typical, not unusual.

Pharmaceutical Biostatistics: Descriptive Statistics