Spread
Range, quartiles, and standard deviation.
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the idea
Spread says how far data wanders from its centre. The range is largest minus smallest. The interquartile range (IQR) is the spread of the middle half: Q3 minus Q1, the medians of the upper and lower halves.
The standard deviation (SD) is the typical distance from the mean. Adding the same number to every value moves the mean but no distances, so the spread stays put. Multiplying every value by a positive k multiplies both the centre and the spread by k.
techniques
Halves, then their middles
- Sort the data and split it into a lower and an upper half.
- Q1 is the median of the lower half, Q3 of the upper half.
- Subtract: Q3 minus Q1.
worked example
Data: 3, 5, 8, 10, 12, 15, 18, 20. Using the median of each half, what is the interquartile range?
- Lower half 3, 5, 8, 10: Q1 = (5 + 8) ÷ 2 = 6.5.
- Upper half 12, 15, 18, 20: Q3 = (15 + 18) ÷ 2 = 16.5.
- 16.5 − 6.5 = 10.
Answer: 10
SD in four steps
- Find the mean.
- Square each value's distance from it.
- Average the squares, dividing by the count. That is the variance.
- Take the square root.
worked example
What is the population standard deviation of 5, 11, 11, 13?
- Mean: 40 ÷ 4 = 10.
- Squared distances: 25, 1, 1, 9, adding to 36.
- 36 ÷ 4 = 9, and the square root of 9 is 3.
Answer: 3
Shift moves, scale stretches
- Adding c: the mean and median go up by c. The range and SD stay the same.
- Multiplying by a positive k: all four are multiplied by k.
worked example
A data set has mean 20 and standard deviation 4. Every value is multiplied by 3, then 5 is added. What is the new standard deviation?
- Multiplying by 3: 4 × 3 = 12.
- Adding 5 moves no gaps, so it stays 12.
Answer: 12
watch out for
- Giving the range when the question asks for the interquartile range.
- Dividing by one less than the count, the sample formula, when the question asks for the population SD.
- Saying the SD goes up when the same number is added to every value. Only the centre moves.
practice
Range & IQR
worked example
Data: 4, 18, 37, 53, 66, 68, 80, 93. Using the median of each half, what is the interquartile range?
Answer: 46.5
- Lower half 4, 18, 37, 53: median 27.5 (Q1).
- Upper half 66, 68, 80, 93: median 74 (Q3).
- IQR = Q3 − Q1 = 74 − 27.5 = 46.5.
Shift & scale
worked example
You add 6 to every value in a data set. What happens to its range?
- increases by 6
- unchanged
- multiplied by 6
- increases by √6
Answer: unchanged
- The largest and smallest values both go up by 6, so the gap between them stays the same.
Standard deviation
worked example
What is the population standard deviation of 59, 59, 69, 69, 71, 75?
Answer: 6
- Mean = 402 ÷ 6 = 67.
- Squared distances from 67: 64, 64, 4, 4, 16, 64, adding to 216.
- 216 ÷ 6 = 36, and √36 = 6.