Recover the data before finding a summary

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You can summarise a frequency table once you know how often each value occurs. Here, recover those frequencies from running totals or two groups. For the basic meanings, revisit mean, median and mode.

A running total already includes earlier values

A frequency tells you how many at this value. A cumulative count tells you how many up to this value.

Value	Cumulative count	New observations at this value
1	2	2
4	5	5 − 2 = 3

The five observations are 1, 1, 4, 4, 4. The second running total includes the first two observations. It does not mean that five separate observations equal 4.

The stacked bar at 4 contains the earlier count of 2 and the new count of 3. It does not show five new observations at 4.

Data table

Category	Value 1	Value 4
Up to 1	2	0
Up to 4	2	3

Recover each new count by subtracting the previous running total. At the first listed value, subtract zero.

The table shows running totals for the number of pets owned by 12 pupils.

Pets	Cumulative count
1	3
2	7
3	8
4	12

Find the median number of pets.

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Recover each frequency
Method
Subtract the previous running total from each one.
Reason
Each running total already includes all the smaller values, so only the increase belongs to the new value.
Working
Answer

Pets Frequency
1 3 − 0 = 3
2 7 − 3 = 4
3 8 − 7 = 1
4 12 − 8 = 4
The frequencies add to 12, the last running total.
Locate the middle positions
Method
With 12 observations, the median is halfway between the 6th and 7th.
Reason
For an even count there is no single middle observation, so the two central positions are used.
Working
Answer
Positions 1–3 have 1 pet and positions 4–7 have 2 pets. So the 6th and 7th observations are both 2.
State the median
Working
Answer
Median = (2 + 2)/2 = 2 pets.

Pets	Frequency
1	3 − 0 = 3
2	7 − 3 = 4
3	8 − 7 = 1
4	12 − 8 = 4

Quick check

The running totals tell you directly that 7 pupils own 2 pets or fewer, so the 6th and 7th pupils, in order, own 2. Taking the middle of the listed values 1, 2, 3, 4 gives 2.5; that ignores how many pupils have each value.

Values 0, 2 and 5 have cumulative counts 2, 7 and 10. Find the separate frequency at 2 and the mean of all ten observations. First subtract consecutive totals, then form a weighted total.

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Separate frequency at 2

Mean

Hint 1

Separate totals from frequencies

Does the cumulative count at 2 include observations of value 0?

Hint 2

Recover then weight the counts

Subtract consecutive cumulative counts for the separate frequencies. Multiply each value by its frequency and divide their sum by the total count.

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Recover the separate frequencies
Method
Subtract the previous cumulative count from each new total.
Reason
Each cumulative count includes all earlier values, so the increase isolates the newly included value.
Working
Answer
At 0: 2. At 2: 7-2 = 5. At 5: 10-7 = 3.
Find the total of the observations
Method
Weight each value by its recovered frequency.
Reason
Repeated observations must contribute once per occurrence, even though the table lists each distinct value only once.
Working
Answer
Total = 0(2) + 2(5) + 5(3) = 25.
Divide by the observation count
Method
Use all ten observations in the denominator.
Reason
The mean is the total shared equally across observations, not across the three distinct values.
Working
Answer
The frequency at 2 is 5 and the mean is 25/10 = 2.5.

Quick check

2 + 5 + 3 = 10 matches the last running total; the mean lies between 0 and 5.

Find the largest increase

The table records how many books were borrowed per visit. Only the listed values are possible. Find every mode of the observations. Separate multiple values with commas.

All possible data values are listed. Each cumulative count includes every observation at or below its value; it is not the frequency of that value alone.
Value	Cumulative count
0	1
2	4
5	7
9	7

All modes (books per visit)

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Books per visit	Separate frequency
0	1 − 0 = 1
2	4 − 1 = 3
5	7 − 4 = 3
9	7 − 7 = 0

The complete mode set is 2 and 5. The running total 7 is neither a book value nor an individual frequency. No visit borrowed 9 books.

As a check, the recovered frequencies add to 7, the final cumulative count.

A flat count adds no observations

Range is largest observed value minus smallest observed value. First decide which listed values actually contribute observations.

This table records temperature changes in °C. Find the range of the observed changes.

All possible data values are listed. Each cumulative count includes every observation at or below its value; it is not the frequency of that value alone.
Value	Cumulative count
−6	0
−2	2
1	5
4	6
9	6

Range (°C)

Locate the observed extremesHide answer

The separate frequencies are 0, 2, 3, 1, 0. No observations equal −6°C or 9°C. The observed values are −2°C, 1°C and 4°C.

Range = 4-(-2) = 6 °C.

Using the first and last labels would give 15°C, but both of those labelled values have zero observations. A running-count table can list a value without any observation at that value.

Take out the same-valued observations

When one group is removed, reconstruct the population you were asked to summarise.

Value	Whole group	Removed group	Remaining group
2	6	4	6 − 4 = 2
5	3	1	3 − 1 = 2

The remaining group is 2, 2, 5, 5. Subtract counts within each row: removing four observations equal to 2 cannot remove an observation equal to 5.

The remaining count also agrees with the group totals: (6 + 3)-(4 + 1) = 4.

Summarise the group that remains

The table records the number of books carried in each bag. Some bags are taken away. Find every mode for the bags that remain. Separate multiple values with commas.

All possible data values are listed. Summarise the observations left after the removed group is taken out of the whole group.
Value	Whole group	Removed group
1	5	4
3	4	1
6	6	3
9	2	2

All modes for the remaining bags (books)

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Books in a bag	Whole-group frequency	Remaining frequency
1	5	5 − 4 = 1
3	4	4 − 1 = 3
6	6	6 − 3 = 3
9	2	2 − 2 = 0

The whole group has only one mode, 6 books. The remaining group has two modes: 3 and 6 books. Using the largest whole-group count would miss the changed population.

Recover the count before the middle

This table records temperature changes in °C. Readings from a separate experiment are removed. Find the median change for the readings that remain.

All possible data values are listed. Summarise the observations left after the removed group is taken out of the whole group.
Value	Whole group	Removed group
0	2	1
2	5	3
6	4	1

Median remaining change (°C)

Order the remaining observationsHide answer

The remaining frequencies are 1, 2 and 3, giving six observations:

0, 2, 2, 6, 6, 6 °C.

Median = (2 + 6)/2 = 4 °C.

The two central positions are different observations, even when a table compresses several equal observations into one row. The median need not itself occur in the data.

Use the recovered data for a different summary

The table records changes in water level, in centimetres. Find their mean. Choose the count and total you need before entering it.

All possible data values are listed. Each cumulative count includes every observation at or below its value; it is not the frequency of that value alone.
Value	Cumulative count
−1	1
0	3
3	5

Mean change (cm)

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The frequencies are 1, 2 and 2, so the five observations are −1, 0, 0, 3 and 3 cm.

n = 1 + 2 + 2 = 5,; T = 1(-1) + 2(0) + 2(3) = 5 cm,; Mean = 5/5 = 1 cm.

Dividing by the three rows would count different values rather than observations. Adding the running totals would count some observations more than once.

Check which information the table retains

These tables list every possible individual value. Their cumulative differences or group differences recover exact frequencies, so the summaries above are exact.

A table of class intervals is different: it does not retain individual values within each class. Midpoints or interpolation can give estimates, but cannot recover the exact original observations.

Values 1, 3 and 7 occur 3, 4 and 3 times in a whole group. A subgroup containing two 1s, one 3 and one 7 is removed. Find the median of the remaining observations. Show how you recover the remaining population.

Write your answer, then compare it with the solution

Your reasoning and answer, or notes from your paper working

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Subtract the subgroup frequencies
Method
Remove each stated subgroup count from its matching value’s frequency.
Reason
Removing four observations changes both the total count and the relative frequency of each value.
Working
Answer
Remaining counts: at 1, 3-2 = 1; at 3, 4-1 = 3; at 7, 3-1 = 2.
Rebuild the ordered remaining data
Method
Write each value the number of times it remains.
Reason
The median depends on ordered positions, so the six remaining observations must be located after removal.
Working
Answer
The remaining list is 1, 3, 3, 3, 7, 7.
Locate the new middle pair
Method
Average the third and fourth values.
Reason
An even count has two central positions; neither the old group nor the removed subgroup determines this median.
Working
Answer
Both central values are 3, so the remaining median is (3 + 3)/2 = 3.

Quick check

The count is 10-4 = 6, agreeing with 1 + 3 + 2. Compare row subtraction, middle positions and the resulting median.

Common mistakes

Reading a cumulative count as a frequency. Subtract the previous running total to find how many are at each value.
Counting a value whose running total did not change. A flat cumulative count means no observations at that value.
Summarising the wrong group. After removing a group, recalculate from the remaining frequencies.
Giving only one mode. List every value with the greatest frequency.

Syllabus and review details

Last reviewed: 2026-11-07

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Guided steps

Reading settings

A running total already includes earlier values

Recover frequencies, then find the median

Recover each frequency

Locate the middle positions

State the median

Recover a missing separate frequency

Recover the separate frequencies

Find the total of the observations

Divide by the observation count

Find the largest increase

Find every modal value

A flat count adds no observations

Locate the observed extremes

Take out the same-valued observations

Summarise the group that remains

Choose the remaining population

Recover the count before the middle

Locate the remaining middle pair

Use the recovered data for a different summary

Choose a count and weighted total

Check which information the table retains

Remove a group before finding its median

Subtract the subgroup frequencies

Rebuild the ordered remaining data

Locate the new middle pair

Common mistakes

Lesson done

A running total already includes earlier values

Recover frequencies, then find the median

Recover a missing separate frequency

Find the largest increase

Find every modal value

A flat count adds no observations

Locate the observed extremes

Take out the same-valued observations

Summarise the group that remains

Choose the remaining population

Recover the count before the middle

Locate the remaining middle pair

Use the recovered data for a different summary

Choose a count and weighted total

Check which information the table retains

Remove a group before finding its median

Common mistakes

Questions