Data Dispersion

1Summary 重點摘要

Range, quartiles, standard deviation, see data spread.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Range — the simplest measureThe range is the simplest measure of dispersion, given by range = maximum − minimum; find the two extreme values and subtract. For example, if the heights of a class run from 150 cm to 180 cm, the range is 180 - 150 = 30 cm. Its advantage is speed, but its weakness is that it is decided by only the largest and smallest values and says nothing about how the middle values are distributed; a single outlier stretches it badly.
2. Step 2: Sort first — must do before quartilesWhether you are finding Q1 or Q3, the data must first be arranged from smallest to largest, and this is the step most often forgotten. For example, the test scores 52, 63, 68, 70, 72, 75, 78, 81, 85, 90 are already in order, so we can move on directly. If you look for quartiles before sorting, the result is bound to be wrong, so always check the order first.
3. Step 3: Median — find the middle firstAfter sorting, find the median, which splits the data into a lower half and an upper half. With an odd number of values the median is the single middle value; with an even number it is the average of the two middle values. For the 10 scores above, the median = (72 + 75) ÷ 2 = 73.5. The lower 5 values are used to find Q1 and the upper 5 to find Q3.
4. Step 4: Q1 and Q3 — quartilesQ1 is the median of the lower half of the data and Q3 is the median of the upper half. In the example: the lower half is 52, 63, 68, 70, 72, whose middle value is 68, so Q1 = 68; the upper half is 75, 78, 81, 85, 90, whose middle value is 81, so Q3 = 81. Once Q1 and Q3 are found, they describe the span of the middle half of the data.
5. Step 5: IQR — spread of the middle halfIQR = Q3 − Q1, the span of the middle half of the data; in the example IQR = 81 − 68 = 13. The great advantage of the IQR is that it depends only on the middle 50% of the data and is completely unaffected by extreme values at either end, which makes it far more robust than the range. If one student scored 0, the range would suddenly become huge, but Q1 and Q3 would hardly move, so the IQR stays stable.
6. Step 6: Standard deviation — most commonThe standard deviation measures how far, on average, each value lies from the mean; the larger it is, the more spread out the data. The steps are: find the mean; subtract the mean from each value and square; add the squared values and divide by the count; finally take the square root. A calculator does the arithmetic in exams, so the key skill is reading it: a standard deviation of 5 marks means values typically lie about 5 marks from the mean.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Range = max - min
• IQR = Q3 - Q1
• Standard deviation measure data spread
• Bigger standard deviation more spread less stable
• Compare two data stability see range and IQR