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2 BIOSTATISTICS TOPIC 5.4 MEASURES OF DISPERSION

3 BIOSTATISTICS TERMINAL OBJECTIVE: 5.4 Calculate Measures of Dispersion.

4 Enabling Objective E.O State the purpose of determining measures of dispersion.

5 Measures of Dispersion Purpose –To describe how much spread there is in a distribution. –Used with a particular measure of central tendency

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7 Enabling Objectives FROM A SET OF STATISTICAL DATA, COMPUTE THE: Range Interquartile range Variance Standard deviation

8 Range Definition – The difference between maximum and minimum.

9 Range uCalculation Arrange data into ascending array Identify the minimum maximum values Calculate the range

10 Interquartile Range Defined: the difference between the 75th percentile (75% of the data) and the 25th percentile (25% of the data) and includes the median, or 50th percentile. Represents the central portion of the normal distribution

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12 Interquartile Range Calculate Interquartile range from individual data –Arrange data in increasing order –Find position of first and third quartiles Q1 = (n+1)/4 Q3 = 3(n+1)/4 = 3xQ1

13 Interquartile Range Calculate Interquartile range from individual data –Arrange data in increasing order –Find position of first and third quartiles Q1 = (n+1)/4 Q3 = 3(n+1)/4 = 3xQ1

14 Interquartile Range –Identify the values –Whole numbers match the observations. – Fractions lie between observations –Interquartile range is Q3-Q1

15 Interquartile Range Example: Observations- 13, 7, 9, 15, 11, 5, 8, 4 STEP 1: Arrange the array 4, 5, 7, 8, 9, 11, 13, 15 STEP 2: Determine Q1 position = (n+1)/4 = (8+1)/4 = 2.25

16 Interquartile Range STEP 3: Count observations from beginning of array 2.25 is the second plus ¼ difference between 2nd and 3rd observations = 5 + ¼(7-5) = 5.5

17 Interquartile Range STEP 4: Determine Q3 position Q3 = 3(n+1)/4 = 3(9)/4 = 6.75 STEP 5: Repeat Step 3 procedure to locate value in array

18 Interquartile Range 6.75 is the sixth plus ¾ difference between the 6th and 7th observation = 11 + ¾(13-11) = 11 + ¾(2) = 12.5

19 Interquartile Range IQR = Q3-Q1 = = 7

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21 Variance Variance (s²) is a measure of dispersion around the mean of a distribution.

22 Variance Calculate Variance from Ungrouped Data –Arrange the data into ascending order

23 Variance Create a frequency distribution table with column headings for X, X, (X- X), (X-X) ². – X = value – X = mean – (X-X) = difference from the mean – (X-X) ² = difference squared

24 Variance Sum the (X-X)² column Formula: (s²) = (X-X)²/n-1 n = total observations

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26 Standard Deviation The standard deviation, s, is the square root of the variance. –s =   (X-X)²/n-1

27 Standard Deviation Indicates how the data falls within the curve of the frequency distribution –Approximately 68% of the values will occur within (+/-) 1 standard deviation (1s) of the mean X. X ± 1s = 68%

28 Standard Deviation –Approximately 95% of the data will occur within (+/-) 2 standard deviations (2s) of the mean X X ± 2s = 95%

29 Standard Deviation – 99.7 % of the data will occur within (+/-) 3 standard deviations (3s) of the mean X ± 3s = 99.7%

30 Standard Deviation – Values which are (+/-) 2s from the mean are only 5% of the total data - a figure that is considered by most researchers to be the cut- off point for "statistical significance."

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33 Enabling Objective E.O State the appropriate measure of dispersion for frequency distributions

34 Choosing Measures Of Dispersion Normal distribution –The standard deviation is preferred Skewed distribution – The interquartile range is preferred

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