Averages and range
Introduced| English | Meaning |
|---|---|
| mean/miːn/ | The sum of a set of values divided by the number of values. |
| median/ˈmiːdiːən/ | The middle value when data are arranged in order, or the mean of the two middle values for an even number of observations. |
| mode/məʊd/ | The value or category occurring most frequently in a data set. |
| range/reɪndʒ/ | The set of output values actually attained by a function on its stated domain. |
| outlier/ˈaʊtlaɪə/ | An observation unusually far from the rest of the data or from the pattern expected under a model. |
What's "normal"?
- The average salary in a company is 50 000 — but most people earn far less, because one CEO earns millions.
- Which average you choose changes the story. Mean 平均数, median 中位数, and mode 众数 each tell a different truth.
Average choice lab
Choose the average that fits the data situation.
The three averages
- Mean $= \dfrac{\text{sum of values}}{\text{how many}}$.
- Median $=$ the middle value when the data is ordered.
- Mode $=$ the most common value.
$4, 7, 7, 2, 5$: ordered $2, 4, 5, 7, 7$. Mean $= \dfrac{25}{5} = 5$; median $= 5$; mode $= 7$.
A box-and-whisker plot shows the five-number summary; the box length is the interquartile range 值域
Find the mean of 4, 7, 7, 2, 5.
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(4 + 7 + 7 + 2 + 5)/5 = 25/5 = 5.
Find the mode of 4, 7, 7, 2, 5.
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7 appears most often.
Find the median of 4, 7, 7, 2, 5.
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Ordered: 2, 4, 5, 7, 7. The middle (3rd) value is 5.
Match each average or measure to how you find it.
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Mean, median and mode are three kinds of average; the range measures spread.
The range
- Range $=$ largest $-$ smallest. It measures spread.
- Range of $4, 7, 7, 2, 5$: $7 - 2 = 5$.
The range is easily distorted. One extreme value (outlier 异常值) makes the range huge, even if most values are close together. The IQR (next lesson) is more robust.
Mean, median, mode and range of a small data set
Find the range of 4, 7, 7, 2, 5.
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Largest − smallest = 7 − 2 = 5.
The range is easily affected by a single extreme value (outlier).
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The range uses only the largest and smallest values, so one outlier can make it very large.
Mean from a frequency table
- When values come with frequencies:
- Values $1, 2, 3$ with frequencies $4, 5, 1$: $\;\dfrac{1(4) + 2(5) + 3(1)}{10} = \dfrac{17}{10} = 1.7$.
Values 1, 2, 3 occur with frequencies 4, 5, 1. Find the mean (1 dp).
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(1·4 + 2·5 + 3·1)/(4+5+1) = 17/10 = 1.7.
Which average to use?
- Mean: uses all the data, but affected by outliers.
- Median: usually less affected by extreme values than the mean — good for skewed data (like salaries).
- Mode: useful for categorical data (most popular colour, most common shoe size).
Ordering data on a number line makes the median obvious — it's the value in the middle.
Even counts and frequency-table medians
- For $2,5,7,10$, the middle pair is 5 and 7, so median $M=(5+7)/2=6$; do not select only one middle value.
- Values 1,2,3 with frequencies 2,3,1 make six observations. Positions 3 and 4 are both 2, so median 2. Frequency counts determine the middle positions, not the number of table rows.
Values 1, 2, 3 have frequencies 2, 3, 1. Find the median.
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Positions 3 and 4 in the six observations are both 2.
You've got it
- mean $=$ total ÷ count; median $=$ middle in order; mode $=$ most common
- range $=$ largest − smallest
- frequency-table mean $= \dfrac{\sum(\text{value} \times \text{freq})}{\sum \text{freq}}$
Write from a mean of 6
- The values are 3, 3, 8 and 10.
- Find the mean.
What is the mean of 3, 3, 8 and 10?
24 divided by 4 is 6.
What is the mean?
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The total 24 shared by 4 values is 6. Other options — 5 (the lesson mean); 3 (the mode of these values).