Statistics

**Three averages.** Eight students scored the following marks out of 20: $$11,\ 14,\ 16,\ 13,\ 18,\ 12,\ 16,\ 10.$$ (a) Find the mean, median, mode, and range. (b) The teacher discovers that one mark was misread and should have been 19 instead of 11. Recompute the four statistics. (c) Which average is most affected by the correction? Why?

**Cumulative frequency analysis.** A class of 80 students sat a test marked out of 50. Here is a cumulative frequency table: | Mark | Cumulative frequency | |------|---------------------| | ≤ 10 | 8 | | ≤ 20 | 28 | | ≤ 30 | 52 | | ≤ 40 | 72 | | ≤ 50 | 80 | Use the curve to find: (a) the median; (b) the lower and upper quartiles, and hence the IQR; (c) the percentage of students scoring more than 35.

**Boxplot comparison.** Two football teams' goal totals over 20 matches are summarised: - **Team A:** min 0, $Q_1 = 1$, median 2, $Q_3 = 3$, max 5. - **Team B:** min 0, $Q_1 = 0$, median 1, $Q_3 = 4$, max 8. Compare the two teams. Which would you back to score more in the next match? Justify.

**Missing values.** A teacher records the test marks for a class of 10 students. Nine of them scored: $$45,\ 67,\ 52,\ 78,\ 84,\ 56,\ 73,\ 65,\ 81.$$ The mean of all 10 students is 68. Find the score of the 10th student.

**Mean from grouped table.** A survey of times taken to complete a puzzle (minutes): | Time (min) | Frequency | |-----------|-----------| | [0, 5) | 8 | | [5, 10) | 14 | | [10, 15) | 22 | | [15, 20) | 12 | | [20, 25) | 4 | (a) Estimate the mean time taken. (b) State the modal class. (c) State the median class.

**Standard deviation (informal).** Two sets of test scores: - Set A: 70, 70, 70, 70, 70 - Set B: 50, 60, 70, 80, 90 (a) Find the mean of each set. (b) Find the range of each set. (c) Which has higher variability? How can you tell visually?

**Skewness.** A boxplot has min 10, $Q_1 = 14$, median 18, $Q_3 = 25$, max 50. (a) Find the IQR. (b) Determine whether the distribution is symmetric, positively skewed, or negatively skewed. (c) Are there any outliers using the $1.5 \times$ IQR rule?

**Combined groups.** Class A has 24 students with mean test score 68. Class B has 16 students with mean 76. (a) Find the combined mean of all 40 students. (b) Is the combined mean the simple average of 68 and 76? Why or why not? (c) If the combined mean were exactly 72, how would the class sizes need to compare?

**Misleading statistic.** A property agent advertises: > "The average price of a flat in our area is £350,000." Sample of 10 recently sold flats: £180k, £200k, £220k, £230k, £240k, £250k, £270k, £280k, £290k, £1,300k. (a) Calculate the mean and median. (b) Comment on which is a fairer summary of "typical" flat prices. (c) What is the role of the outlier here?

**Reverse problem.** The mean of 5 distinct positive integers is 10, the median is 9, and the mode is 7. (a) Find a possible set of 5 integers. (b) How many different sets are possible? Find all.

**Effect of transformation.** A data set has mean 60 and IQR 12. (a) Each value is increased by 5. What is the new mean and IQR? (b) Each value is doubled. What is the new mean and IQR? (c) Each value is increased by 5 and then doubled. What is the new mean and IQR?

**Survey design.** A council wants to know the average household income in a town. They survey 50 randomly selected households. (a) Why might the mean of the 50 incomes not equal the true population mean? (b) Why might the median be a better statistic than the mean? (c) How could they improve the estimate?