Statistics

Measures of central tendency, such as the mean, median and mode, give a single value representing a set of data, but they do not show how widely the observations are spread. Two classes may have the same average mark but very different spreads. This chapter studies measures of dispersion, which describe this scatter. The simplest is the range, the difference between the largest and smallest observations, but it depends only on two extreme values. The mean deviation about the mean or median uses every observation: it is the average of the absolute deviations from that central value. You will learn to calculate the mean deviation for ungrouped data, discrete frequency distributions and continuous frequency distributions using class marks. The chapter then introduces the variance, the mean of the squared deviations from the mean, and the standard deviation, its positive square root. You will use the shortcut formula variance = (sum of x^2)/n - (mean)^2 and the step-deviation method, and study how adding a constant or multiplying by a constant affects the mean and variance.

Detailed Worksheet: Statistics Section A - Definitions (10 marks) 1. Find the range of 6, 7, 10, 12, 13, 4, 8, 12. (2 marks) 2. Define mean deviation about the mean. (2 marks) 3. Find the variance of 1, 2, 3, 4, 5. (2 marks) 4. State how the standard deviation changes if every observation is increased by 5. (2 marks) 5. Why is the range not always a good measure of dispersion? (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the mean deviation about the mean of 6, 7, 10, 12, 13, 4, 8, 12. (3 marks) 7. Find the mean deviation about the median of 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21. (3 marks) 8. Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9. (3 marks) 9. The mean and variance of n observations are 10 and 4. Each observation is multiplied by 3. Find the new mean and variance. (3 marks) 10. Find the variance of the first 10 natural numbers using the formula (n^2 - 1)/12, and check it by direct calculation of the mean. (3 marks) Section C - Diagrams (10 marks) 11. Draw a dot plot of the data 40, 50, 60, 70, 80 and mark the mean and the deviations of each observation from it. (4 marks) 12. Draw a histogram for the distribution with classes 0-10, 10-20, 20-30, 30-40, 40-50 and frequencies 5, 8, 15, 16, 6. (3 marks) 13. Draw two dot plots with the same mean but different spreads, and explain which has the larger standard deviation. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Find the mean deviation about the mean for the data: x = 2, 5, 6, 8, 10, 12 with frequencies f = 2, 8, 10, 7, 8, 5. (5 marks) 15. Find the mean, variance and standard deviation of: x = 4, 8, 11, 17, 20, 24, 32 with frequencies f = 3, 5, 9, 5, 4, 3, 1. (5 marks) 16. Find the mean, variance and standard deviation for classes 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 5, 8, 15, 16, 6. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show the table of deviations for every calculation. Round to two decimal places.