The Mean Sits Exactly Where the Data Balances
Learn why the distances above and below the mean always cancel, predict how the mean moves when a new value is added, see how the median responds instead, and compute both from a frequency table.
Why do the gaps above and below the mean always cancel out?
Because the mean is the point where the data balances. Add up how far every value sits above it and how far every value sits below it, and the two totals are always equal.
For the mean is , and the distances below total while those above total too. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: the mean as a balance point, how it moves, how the median responds, and both from a frequency table.
For the mean is , and the distances below total while those above total too. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: the mean as a balance point, how it moves, how the median responds, and both from a frequency table.
Formula
How do you show that the distances above and below the mean are equal?
Calculate the mean, then find each value's deviation from it — the value minus the mean — and add the negatives and positives separately.
Worked example. For :
Now the deviations:
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The below total is , and the above total is . Equal, so the deviations sum to zero:
A second check on , whose mean is : deviations are , adding to .
The fair share reading explains why. If the five people pooled their amounts and split them equally, everyone would get — so whatever the ones above the mean give up is exactly what the ones below receive.
The picture worth keeping is a see-saw. Place the values along a line and the mean is the pivot where it balances, which is also why a single very large value pulls the mean strongly towards itself — it sits far from the pivot and needs many values on the other side to hold it.
Worked example. For :
Now the deviations:
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-
-
-
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The below total is , and the above total is . Equal, so the deviations sum to zero:
A second check on , whose mean is : deviations are , adding to .
The fair share reading explains why. If the five people pooled their amounts and split them equally, everyone would get — so whatever the ones above the mean give up is exactly what the ones below receive.
The picture worth keeping is a see-saw. Place the values along a line and the mean is the pivot where it balances, which is also why a single very large value pulls the mean strongly towards itself — it sits far from the pivot and needs many values on the other side to hold it.
How does the mean change when you add a new value?
Compare the new value with the current mean:
- New value greater than the mean — the mean rises
- New value less than the mean — the mean falls
- New value equal to the mean — the mean is unchanged
Worked examples, starting from with mean and sum .
Adding 30, which is greater than 18:
The mean rose from to , as predicted.
Adding 6, which is less than 18:
It fell.
Adding 18, exactly the mean:
Unchanged.
Finding a value to hit a target mean. What must be added to make the mean ? Six values with mean need a total of , so the value is
Finding a missing observation. The mean of six numbers is and five of them are , summing to . The required total is , so the missing value is .
Predicting the direction before calculating is the skill being tested, and it is a free check on the arithmetic. Adding 30 to a set with mean 18 must push the mean up — so an answer below 18 would be wrong without any further examination.
- New value greater than the mean — the mean rises
- New value less than the mean — the mean falls
- New value equal to the mean — the mean is unchanged
Worked examples, starting from with mean and sum .
Adding 30, which is greater than 18:
The mean rose from to , as predicted.
Adding 6, which is less than 18:
It fell.
Adding 18, exactly the mean:
Unchanged.
Finding a value to hit a target mean. What must be added to make the mean ? Six values with mean need a total of , so the value is
Finding a missing observation. The mean of six numbers is and five of them are , summing to . The required total is , so the missing value is .
Predicting the direction before calculating is the skill being tested, and it is a free check on the arithmetic. Adding 30 to a set with mean 18 must push the mean up — so an answer below 18 would be wrong without any further examination.
How does the median change when values are inserted?
The median is the middle value of the ordered data, so it responds to position rather than to size — and it usually moves far less than the mean.
For an odd count the median is the middle value, at position . For an even count it is the mean of the two middle values.
Worked examples, starting from with median .
Insert 30. The ordered set becomes — six values, so the median is
It moved only from 18 to 19, while the mean moved from 18 to 20.
Insert 6. The set becomes , so the median is
Insert a very large value instead. Replace 30 with : the ordered set is and the median is still
but the mean jumps to .
That last comparison is the whole point of having two averages. The median barely noticed the , because it only asks where a value sits in the order; the mean was dragged to , because it adds every value at full size.
So an outlier distorts the mean and leaves the median almost untouched — which is why the median is preferred for describing incomes or house prices, where a few very large values would otherwise misrepresent a typical case.
For an odd count the median is the middle value, at position . For an even count it is the mean of the two middle values.
Worked examples, starting from with median .
Insert 30. The ordered set becomes — six values, so the median is
It moved only from 18 to 19, while the mean moved from 18 to 20.
Insert 6. The set becomes , so the median is
Insert a very large value instead. Replace 30 with : the ordered set is and the median is still
but the mean jumps to .
That last comparison is the whole point of having two averages. The median barely noticed the , because it only asks where a value sits in the order; the mean was dragged to , because it adds every value at full size.
So an outlier distorts the mean and leaves the median almost untouched — which is why the median is preferred for describing incomes or house prices, where a few very large values would otherwise misrepresent a typical case.
How do you find the mean and median from a frequency table?
Multiply each value by its frequency for the mean, and use the cumulative frequency to locate the median.
Worked example. Marks scored by 10 students:
- Mark , frequency
- Mark , frequency
- Mark , frequency
- Mark , frequency
The mean. Multiply and add:
The total frequency is , so
The median. There are values, so the median is the mean of the 5th and 6th. Build up the positions:
- The two s occupy positions and
- The three s occupy positions
- The four s occupy positions
- The single occupies position
So the 5th value is and the 6th is , giving
The mode here is , the value with the highest frequency.
The step to get right is multiplying by the frequency, and skipping it is the standard error. Adding just the four distinct marks gives , which is not the mean — each mark must be counted as many times as it actually occurred, which is what the frequency records.
Worked example. Marks scored by 10 students:
- Mark , frequency
- Mark , frequency
- Mark , frequency
- Mark , frequency
The mean. Multiply and add:
The total frequency is , so
The median. There are values, so the median is the mean of the 5th and 6th. Build up the positions:
- The two s occupy positions and
- The three s occupy positions
- The four s occupy positions
- The single occupies position
So the 5th value is and the 6th is , giving
The mode here is , the value with the highest frequency.
The step to get right is multiplying by the frequency, and skipping it is the standard error. Adding just the four distinct marks gives , which is not the mean — each mark must be counted as many times as it actually occurred, which is what the frequency records.
Exam tip
Exam tip: predicting the direction before you calculate
Data questions come with free checks, so use them.
Before computing a new mean, predict the direction: a value above the old mean raises it, below lowers it, equal leaves it. Then confirm your answer agrees.
For a missing observation, show the two lines: *required total , then missing *.
Order the data before finding any median, and with an even count take the mean of the two middle values.
From a frequency table, multiply value by frequency before adding, and divide by the total frequency — not by the number of distinct values.
And when asked which average suits the data, name the outlier and say the median is less affected by it. That comparison is what the mark is for.
Before computing a new mean, predict the direction: a value above the old mean raises it, below lowers it, equal leaves it. Then confirm your answer agrees.
For a missing observation, show the two lines: *required total , then missing *.
Order the data before finding any median, and with an even count take the mean of the two middle values.
From a frequency table, multiply value by frequency before adding, and divide by the total frequency — not by the number of distinct values.
And when asked which average suits the data, name the outlier and say the median is less affected by it. That comparison is what the mark is for.
Did you know
Why does one huge value move the mean but not the median?
Because the two averages ask different questions.
The mean adds every value at its full size, so a value of 300 among numbers near 20 contributes 300 to the total and hauls the average up with it. The median only asks where each value sits in the queue — and 300 is simply the last one, exactly as 30 would have been.
Change that 300 to 3000 and the median does not move at all, while the mean climbs again. Two averages built from the same data can therefore tell quite different stories, which is why a report quoting only one of them is worth questioning.
The mean adds every value at its full size, so a value of 300 among numbers near 20 contributes 300 to the total and hauls the average up with it. The median only asks where each value sits in the queue — and 300 is simply the last one, exactly as 30 would have been.
Change that 300 to 3000 and the median does not move at all, while the mean climbs again. Two averages built from the same data can therefore tell quite different stories, which is why a report quoting only one of them is worth questioning.
Key takeaways
Mean, median and frequency tables: quick revision
- , and the deviations from the mean always add to zero — for the mean is 18 with 9 below and 9 above.
- The mean is the balance point of the data, like the pivot of a see-saw.
- Adding a value above the mean raises it, below lowers it, equal leaves it — adding 30 to that set gives a mean of 20.
- For a target mean, find the required total: six values averaging 19 need 114, so add 24.
- The median is the middle of the ordered data, or the mean of the two middle values for an even count — inserting 30 moved it only from 18 to 19, and inserting 300 moves it not at all.
- From a frequency table, the mean is , and the median comes from cumulative positions, giving here.
You will remember all of this far better after answering five questions on it than after reading it twice.
- The mean is the balance point of the data, like the pivot of a see-saw.
- Adding a value above the mean raises it, below lowers it, equal leaves it — adding 30 to that set gives a mean of 20.
- For a target mean, find the required total: six values averaging 19 need 114, so add 24.
- The median is the middle of the ordered data, or the mean of the two middle values for an even count — inserting 30 moved it only from 18 to 19, and inserting 300 moves it not at all.
- From a frequency table, the mean is , and the median comes from cumulative positions, giving here.
You will remember all of this far better after answering five questions on it than after reading it twice.