Measures of Spread Revision
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This document defines and explains various measures of spread for data sets, including range, interquartile range, mean deviation, variance, and standard deviation. It provides formulas to calculate each measure from discrete data sets and frequency tables. Examples are given to demonstrate calculating measures such as standard deviation, variance, range, interquartile range, first and third quartiles, and median.
Measures of Spread Revision
- 1. Measures of Spread Range: -The range is the highest value minus the lowest value. The interquartile range is the third quartile minus the first quartile Mean deviation:- This tells us on the average, how far all the values are from the center. Variance: -The variance is the average of the squares of the distance each value is from the
- 2. Standard deviation: 1. Use 𝑆 = 𝑥− 𝑥 2 𝑛 , if the data set is in discrete form 2. Use 𝑆 = 𝑓𝒙 𝟐 𝑓 − 𝑓𝒙 𝑓 2 , if the data set is in frequency table 3. Use 𝑆 = 𝑓𝒅 𝟐 𝑓 − 𝑓𝒅 𝑓 2 , if you know the assumed mean
- 3. 1. Mean deviation = |𝑿− 𝑿| 𝑛 , if the data set is in discrete form 2. 𝑀𝑒𝑎𝑛 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑓|𝑿− 𝑿| 𝑓 , if the data set is in frequency table MEAN DEVIATION
- 4. The mean of 2 − 𝑡, 4 + 𝑡, 3 − 2𝑡, 2 + 𝑡 and 𝑡 − 1 is. JAMB 2018 SOLUTION Mean, 𝑋 = 𝑥 𝑛 = 2 − 𝑡 + 4 + 𝑡 + 3 − 2𝑡 + 2 + 𝑡 + (𝑡 − 1) 5 2 + 4 + 3 + 2 − 1 − 𝑡 + 𝑡 − 2𝑡 + 𝑡 + 𝑡 5 10 + 0 5 10 2 = 2
- 5. The distribution of scores in a class test are as follows: 2,8,6,5,8,6,6,5,6 and 5. Find the product of the modal and the median score JAMB 2018 SOLUTION 6 occurs most Modal mark= 6 For median mark, we arrange the numbers in order: 2,5,5,5,6,6,6,6,8,8 Median mark = 6+_6 2 12 2 = 6 Their product = 6 × 6 = 36
- 6. 1. The standard deviation of the data set 21,15, 𝑥, 𝑎𝑛𝑑 18 is 5 3 4 . Find the value of 𝑥. A. 15 B. 16 C. 17 D. 19 E. 20 The correct answer is 17
- 7. 2. Compute the variance of 21,15,22, 𝑎𝑛𝑑 18. A. 3.5 B. 4.5 C. 5.5 D. 6.5 E. 7.5 The correct answer is 7.5
- 8. 3. The table below is a distribution of marks scored by 20 students out 10. Calculate the range A. 4 B. 5 C. 6 D. 7 E. 8 The range is 4 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟒 𝟓 𝟔 𝟕 𝟖 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 6 3 4 3 4
- 9. 4. The table below is a distribution of marks scored by 20 students out 10. Calculate the standard deviation correct to 2 decimal places? A. 1.50 B. 2.50 C. 3.50 D. 4.50 E. 5.50 The correct answer is 1.50 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟒 𝟓 𝟔 𝟕 𝟖 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 6 3 4 3 4
- 10. 5. The histogram represents the distribution of marks scored by students in a test. Calculate the variance of the distribution. A. 1.25 B. 2.25 C. 3.25 D. 4.25 E. 5.25 The correct answer is 2.25
- 11. 6. The table below is a distribution of marks scored by 20 students out 10. Calculate the inter-quartile range? A. 1.0 B. 2.0 C. 3.0 D. 4.0 E. 5.0 The correct answer is 3.0 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟒 𝟓 𝟔 𝟕 𝟖 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 6 3 4 3 4
- 12. 7. The histogram represents the distribution of marks scored by students in a test. Calculate the semi interquartile range. A. 1.5 B. 2.5 C. 3.5 D. 4.5 E. 5.5 The correct answer is 1.5
- 13. 8. The table below is a distribution of marks scored by students in a test. Calculate the third quartile? A. 1.50 B. 2.50 C. 3.50 D. 4.50 E. 5.50 The correct answer is 4.50 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 𝟐 𝟑 𝟒 𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 3 2 4 3 4
- 14. 9. The table below is a distribution of marks scored by students in a test. Calculate the first quartile? A. 1.0 B. 2.0 C. 3.0 D. 4.0 E. 5.0 The correct answer is 2.0 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 𝟐 𝟑 𝟒 𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 3 2 4 3 4
- 15. 10. The table below is a distribution of marks scored by students in a test. Calculate the inter-quartile range? A. 1.5 B. 2.5 C. 3.5 D. 4.5 E. 5.5 The correct answer is 2.0 𝑴𝒂𝒓𝒌𝒔 (𝒙) 𝟏 𝟐 𝟑 𝟒 𝟓 𝑁𝑜. 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 3 2 4 3 4
- 16. Summary 1. The first quartile is in 1 4 𝑓 𝑡ℎ position 2. The median is in 1 2 𝑓 𝑡ℎ position 3. The third quartile is in 3 4 𝑓 𝑡ℎ position
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