C10.4; E10.4 - Mean, median and mode
Two groups can have the same total but quite different typical values. C10.4 and E10.4 develop the mean, median and mode for lists and ungrouped frequency tables, then distinguish what each average tells you. These hand methods use exact individual values rather than grouped intervals.
Mean
Four jars contain 6, 8, 8 and 10 counters. If the counters were shared equally, how many would each jar contain? Their total is 32, so each would receive 8. That equal share is the mean; the original jars need not have had equal contents.
Mean
The mean is the sum of all the observations divided by the number of observations. It uses every value, including repeats and zeros.
For a list, add the values. In a frequency table, a value occurring several times contributes that value multiplied by its frequency. A zero observation contributes nothing to the total amount, but still contributes one to the number of observations.
Mean from individual values or exact frequencies
Use the frequency form when rows give exact values. These average rules are not supplied on the formula lists: learn them. Grouped intervals do not give exact values and require separate treatment.
This equal-sharing interpretation explains the division: the total must be split across observations, not across the number of different values.
Worked example 1: a list with repeats
Question. Find the mean of .
Plan. Add all four observations, including both eights, and divide by four.
Answer. 8 counters per jar.
Check. , so replacing each count by the mean preserves the total. The mean also lies between the smallest and greatest counts, 6 and 10.
The next data are compressed into a table. Multiplication recreates the same repeated addition without writing out every observation.
Worked example 2: exact frequencies including zero
Question. Ten pupils report numbers of pets as follows. Find the mean number of pets.
| Number of pets | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Frequency | 2 | 3 | 4 | 1 |
Plan. Multiply each number of pets by its frequency to count all pets, then divide by all pupils.
Answer. 1.4 pets per pupil.
Check. , matching the pet total. A mean need not be a possible individual count: no pupil has to own 1.4 pets.
If the mean and count are given instead, multiplying them recovers the total. For example, six values with mean 7 have total . This reversal is useful when one value is missing.
Watch out
Dividing 14 by the four rows would give 3.5, which is wrong: the table represents ten pupils, not four. Keep zero-pet pupils in the denominator.
Use both rows of the new table.
Median
The median answers a different question: where is the middle of the ordered observations? Putting values in order matters. A middle entry in an unsorted list has no special statistical meaning.
Median
The median is the middle value when observations are ordered. With an odd number of observations there is one middle value. With an even number, take the mean of the two middle values.
For odd, the middle position is . For even, use positions and . These are positions in the ordered data, not data values. In an ungrouped frequency table, running frequency totals locate these positions without writing every repeat. This is a counting method on exact values, rather than a cumulative-frequency graph.
Worked example 3: an odd list
Question. Find the median of .
Plan. Order the five values, then identify the third.
Answer. The median is 7.
Check. Two values lie before 7 and two after it. The original third entry, 6, would not split the ordered list equally.
Change to an even number of observations given as frequencies. The two middle positions can fall in different rows, so read both.
Continue this lesson.
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