These eight questions cover the four lessons in dispersion of ungrouped data. Work each one on paper first.
Questions
Question 1. Find the range of 14, 9, 21, 17, 12, 25 (1 mark).
Answer
The largest is 25 and the smallest is 9, so the range is 25 − 9 = 16.
Question 2. Find the interquartile range of 14, 9, 21, 17, 12, 25 (3 marks).
Answer
Sorted: 9, 12, 14, 17, 21, 25. The median is (14 + 17) ÷ 2 = 15.5. Lower half 9, 12, 14 gives Q1 = 12. Upper half 17, 21, 25 gives Q3 = 21.
Interquartile range = 21 − 12 = 9.
Question 3. Find the mean and variance of 3, 5, 7, 9, 11 (3 marks).
Answer
Mean = 35 ÷ 5 = 7. Deviations: −4, −2, 0, 2, 4. Squared: 16, 4, 0, 4, 16, total 40. Variance = 40 ÷ 5 = 8.
Question 4. Find the standard deviation of the data in Question 3 (1 mark).
Answer
σ = √8 = 2.83 (to two decimal places).
Question 5. The data 1, 4, 4, 5, 6 has Σx² = 94. Use the Σx² method to find the variance (3 marks).
Answer
Mean = 20 ÷ 5 = 4. Variance = 94 ÷ 5 − 4² = 18.8 − 16 = 2.8.
Check with deviations: −3, 0, 0, 1, 2 squared 9, 0, 0, 1, 4, total 14, and 14 ÷ 5 = 2.8.
Question 6. Team P scored 8, 10, 12, 14, 16 and Team Q scored 11, 12, 12, 12, 13. Compare them (4 marks).
Answer
Both means are 12. For P the variance is (16 + 4 + 0 + 4 + 16) ÷ 5 = 8, so σ = 2.83. For Q the variance is (1 + 0 + 0 + 0 + 1) ÷ 5 = 0.4, so σ = 0.63.
The means are equal, and the standard deviation of Q is smaller, so Team Q is more consistent.
Question 7. The data 5, 7, 9, 11, 13 has mean 9 and variance 8. Each value is multiplied by 4, then 3 is added. Find the new mean and variance (3 marks).
Answer
New mean = 9 × 4 + 3 = 39. The addition does not change the spread, and the multiplication multiplies the variance by 4² = 16.
New variance = 8 × 16 = 128.
Question 8. The data 6, 8, 10, 12, 14 has mean 10. A new value of 10 is added. State what happens to the mean and to the standard deviation, with a reason (3 marks).
Answer
The mean stays at 10, because the new value equals the mean. The sum of squared deviations stays 40, but it is now divided by 6 instead of 5.
So the variance falls from 8 to 6.67, and the standard deviation decreases from 2.83 to 2.58.
If you got these wrong
Questions 1 and 2 use calculating range and interquartile range. Questions 3 to 5 use finding variance and standard deviation.
Question 6 uses comparing data sets using centre and spread. Questions 7 and 8 use predicting the effect of changing a data set.
Build a timed mixed set with the timed original practice session builder. For a teacher to mark your working, see online one-to-one Mathematics tuition.