When the Average Lies and the Middle Value Tells the Truth
Learn to calculate the mean, recover a missing observation from it, find the median for odd and even data sets, and judge which measure describes your data honestly.
What is the mean of a set of data?
The mean, or arithmetic average, is the total of all the observations divided by how many there are. It answers the question: if the whole quantity were shared out equally, how much would each one get?
This page covers everything in the ICSE Class 6 Mathematics chapter on mean and median of ungrouped data: calculating the mean, working backwards to a missing value, finding the median, and deciding which measure actually describes a data set fairly.
This page covers everything in the ICSE Class 6 Mathematics chapter on mean and median of ungrouped data: calculating the mean, working backwards to a missing value, finding the median, and deciding which measure actually describes a data set fairly.
Formula
How do you calculate the mean?
The formula is:
Worked example. Five students score 12, 15, 18, 10 and 20 marks.
A second example with units. Four bags weigh 2.5 kg, 3 kg, 4.5 kg and 2 kg, so the total is 12 kg and the mean is kg.
In real life this is how a shopkeeper finds average daily sales, or a teacher finds the class average.
The mean need not be one of the original values, and it need not even be a whole number. Three children with 1, 2 and 2 sweets have a mean of , about 1.67 sweets — a figure nobody actually holds.
Worked example. Five students score 12, 15, 18, 10 and 20 marks.
A second example with units. Four bags weigh 2.5 kg, 3 kg, 4.5 kg and 2 kg, so the total is 12 kg and the mean is kg.
In real life this is how a shopkeeper finds average daily sales, or a teacher finds the class average.
The mean need not be one of the original values, and it need not even be a whole number. Three children with 1, 2 and 2 sweets have a mean of , about 1.67 sweets — a figure nobody actually holds.
Formula
How do you find a missing observation from the mean?
Rearranging the mean formula gives the total, which is the key to working backwards:
Worked example. The mean of five numbers is 14, and four of them are 10, 12, 18 and 15. Find the fifth.
Check it: , and . Correct.
The same rearrangement answers "what must I score next time" questions. If your mean over four tests is 72 and you want a mean of 75 over five, you need a total of against a current total of , so the fifth score must be 87.
Worked example. The mean of five numbers is 14, and four of them are 10, 12, 18 and 15. Find the fifth.
Check it: , and . Correct.
The same rearrangement answers "what must I score next time" questions. If your mean over four tests is 72 and you want a mean of 75 over five, you need a total of against a current total of , so the fifth score must be 87.
How do you find the median for odd and even data sets?
The median is the middle value once the data is arranged in ascending order. Sorting first is not optional — an unsorted list has no meaningful middle.
With an odd number of observations there is one middle value. For 7, 3, 9, 5, 11, sorted to 3, 5, 7, 9, 11, the median is the third value, 7.
With an even number there are two middle values, so the median is their mean. For 4, 8, 6, 10, sorted to 4, 6, 8, 10, the two middle values are 6 and 8, so
For observations, the median sits at position when is odd, and between positions and when is even.
As with the mean, the median of an even set need not be one of the original values — here 7 does not appear in the data at all.
With an odd number of observations there is one middle value. For 7, 3, 9, 5, 11, sorted to 3, 5, 7, 9, 11, the median is the third value, 7.
With an even number there are two middle values, so the median is their mean. For 4, 8, 6, 10, sorted to 4, 6, 8, 10, the two middle values are 6 and 8, so
For observations, the median sits at position when is odd, and between positions and when is even.
As with the mean, the median of an even set need not be one of the original values — here 7 does not appear in the data at all.
When does the median describe data better than the mean?
The mean uses every value, so a single extreme observation drags it away from the bulk of the data. The median only cares about position, so it stays put.
Consider five daily wages: Rs 300, 320, 340, 350 and Rs 2,000.
The median is the middle value, Rs 340. Four of the five workers earn nowhere near Rs 662, so the median describes this group far more honestly — the mean has been pulled up by one unusually high wage.
With evenly spread data the two measures sit close together, and then either is fine. For marks of 12, 15, 18, 10, 20 the mean is 15 and the median is also 15.
So the choice is not arbitrary: use the mean for evenly spread data, and prefer the median when one or two values are far from the rest.
Consider five daily wages: Rs 300, 320, 340, 350 and Rs 2,000.
The median is the middle value, Rs 340. Four of the five workers earn nowhere near Rs 662, so the median describes this group far more honestly — the mean has been pulled up by one unusually high wage.
With evenly spread data the two measures sit close together, and then either is fine. For marks of 12, 15, 18, 10, 20 the mean is 15 and the median is also 15.
So the choice is not arbitrary: use the mean for evenly spread data, and prefer the median when one or two values are far from the rest.
Exam tip
The mistake most students make with the median
The single biggest error is taking the middle of the list as given, without sorting it first.
For 7, 3, 9, 5, 11 the middle of the unsorted list is 9, but the median is 7. The number in the middle position means nothing until the data is in order.
Build a two-step habit: write the sorted list out as a separate line, then count to the middle. For an even count, remember you need the mean of two middle values, not just the lower one — and write out that small division rather than doing it in your head, since is where careless answers of 6 or 8 appear.
For 7, 3, 9, 5, 11 the middle of the unsorted list is 9, but the median is 7. The number in the middle position means nothing until the data is in order.
Build a two-step habit: write the sorted list out as a separate line, then count to the middle. For an even count, remember you need the mean of two middle values, not just the lower one — and write out that small division rather than doing it in your head, since is where careless answers of 6 or 8 appear.
Did you know
Why can the mean be a value that nobody actually has?
The mean describes the data set as a whole, not any individual in it. It answers "what would each get if the total were shared equally", and equal shares rarely match what anyone really had.
That is why a statement like "the average family has 2.3 children" is perfectly correct arithmetic while describing no family at all — the 0.3 belongs to the total, not to a child.
That is why a statement like "the average family has 2.3 children" is perfectly correct arithmetic while describing no family at all — the 0.3 belongs to the total, not to a child.
Key takeaways
Mean and median in 30 seconds
- The mean is the sum of the observations divided by their number, and need not be one of the values or a whole number.
- Rearranged, sum = mean × number of observations, which recovers a missing value or a required next score.
- The median is the middle value of data sorted in ascending order — always sort before counting to the middle.
- For an even number of observations, the median is the mean of the two middle values.
- One extreme value pulls the mean but not the median, so prefer the median when the data contains an outlier and the mean when it is evenly spread.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Rearranged, sum = mean × number of observations, which recovers a missing value or a required next score.
- The median is the middle value of data sorted in ascending order — always sort before counting to the middle.
- For an even number of observations, the median is the mean of the two middle values.
- One extreme value pulls the mean but not the median, so prefer the median when the data contains an outlier and the mean when it is evenly spread.
You will remember all of this far better after answering five questions on it than after reading it twice.