Averages, and which to use
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. |
| symmetric/sɪˈmetrɪk/ | Having corresponding parts or relationships that remain equivalent under a specified transformation or interchange. |
| skewed/skjuːd/ | skewed |
A centre answers a specified question
- The mean 平均数 measures total per observation; the median 中位数 locates the ordered middle; the mode 众数 identifies a most frequent value.
- No single summary answers every question. A mode can describe a category, while mean and median need a suitable numerical or ordered scale.
Combine totals before combining groups
- Compute the mean as $\bar x=\sum x/n$, or $\sum fx/\sum f$ for exact-value frequency data. For an even count, the numerical median averages the middle two ordered values.
- To combine group means, weight them by group size. Ten scores with mean 60 and twenty with mean 75 have combined mean 70, not 67.5.
Which average can be used for favourite subject?
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Categories cannot be added or ordered meaningfully, so only the most frequent value applies.
Interpret shape without inventing observations
- A symmetric 对称 distribution need not have its mode at the centre if it has two peaks. Do not assume mean, median and mode always coincide.
- A skewed 偏态 pattern or extreme values can separate the mean from the median. Inspect the supplied data rather than treating one gap between summaries as a complete shape description.
A mean much larger than the median suggests a few unusually large values.
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A mean well above the median can suggest high extreme values or a right-skewed pattern. It does not uniquely determine the full data shape; inspect the observations.
Nine incomes of 30000 and one of 150000. The mean is $\bar x=[9(30000)+150000]/10=42000$, while median is 30000. The mean correctly describes total income per person; it does not claim anybody earns 42000. The median identifies the ordered middle.
Nine people earn 30000 and one earns 150000. What is the median?
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Ordered, the middle values are both 30000. The single large value does not move it.
Which average best describes a typical income in that group?
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In this supplied group, nine of ten incomes equal the median 30000. The mean 42000 is still a valid total-per-person summary, but is not what most individuals earn.
Explain what your summary represents. In the supplied income example the median also equals what nine people earn. That particular fact comes from the actual list, not from a general definition of median. Sheet 2.6 compares weighted means and the effect of a changed extreme.
A grouped frequency table with numeric intervals usually loses individual values. Using class midpoints gives an estimated mean unless further information makes it exact. Do not confuse interval groups with a frequency table of exact distinct values, whose mean can be calculated exactly.
What can generally be calculated from a grouped frequency table of numeric intervals when the individual values are not supplied?
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Class midpoints give an estimated mean unless additional information makes it exact. A table of exact distinct values and their frequencies is different and permits an exact mean.