BIOSTATISTICS

2. BIOSTATISTICS TOPIC 5.4 MEASURES OF DISPERSION

3. BIOSTATISTICS • TERMINAL OBJECTIVE: • 5.4 Calculate Measures of Dispersion.

4. Enabling Objective • E.O. 5.4.1 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

7. Enabling Objectives FROM A SET OF STATISTICAL DATA, COMPUTE THE: 5.4.2 Range. 5.4.3 Interquartile range. 5.4.4 Variance. 5.4.5 Standard deviation

8. Range • Definition • The difference between maximum and minimum.

9. Range • Calculation • 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

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 = 12.5 - 5.5 = 7

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

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."

33. Enabling Objective E.O. 5.4.6 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