One Wrong Entry Can Shift a Mean but Not a Median
Find the mean of raw data and of a frequency distribution, locate the median for an odd or even number of observations, and recover a missing value or correct a mean after a misread entry.
Why can the mean and the median of the same data be so different?
Five shopkeepers in a lane report their daily earnings, in rupees:
The mean is
The median — the middle value once they are in order — is .
One of those numbers describes the lane and the other does not. Four of the five shopkeepers earn or less, so a typical earning of is nonsense; is honest. The single large value dragged the mean upward and left the median untouched.
That is the difference in one line. The mean uses every value's size, so one extreme entry moves it. The median uses only position, so changing to would send the mean past and leave the median at exactly.
Neither is right in general. The mean is the better summary when the data is evenly spread, and it is the one that can be recombined — you can recover a total from it. The median is better when a few extreme values would mislead, which is why incomes, house prices and waiting times are usually reported as medians.
This page covers the ICSE Class 9 Mathematics chapter on mean and median: the mean of raw and arrayed data, the mean of a frequency distribution, the median for odd and even counts, and questions on missing or misread observations.
The mean is
The median — the middle value once they are in order — is .
One of those numbers describes the lane and the other does not. Four of the five shopkeepers earn or less, so a typical earning of is nonsense; is honest. The single large value dragged the mean upward and left the median untouched.
That is the difference in one line. The mean uses every value's size, so one extreme entry moves it. The median uses only position, so changing to would send the mean past and leave the median at exactly.
Neither is right in general. The mean is the better summary when the data is evenly spread, and it is the one that can be recombined — you can recover a total from it. The median is better when a few extreme values would mislead, which is why incomes, house prices and waiting times are usually reported as medians.
This page covers the ICSE Class 9 Mathematics chapter on mean and median: the mean of raw and arrayed data, the mean of a frequency distribution, the median for odd and even counts, and questions on missing or misread observations.
How do you find the mean of a simple list of numbers?
Add the values and divide by how many there are.
Worked example 1. Find the mean of .
Worked example 2 — with units. The heights of five students are cm, cm, cm, cm and cm. Find the mean height.
The check that costs nothing: a mean must lie between the smallest and the largest value. Here , so the answer is at least possible. A mean outside that range is always an arithmetic slip, usually a missed value in the sum.
Worked example 3 — the assumed-mean shortcut. Find the mean of without adding five three-digit numbers.
Take an assumed mean and work with the deviations :
Check directly: , and , as required.
Why the shortcut is exact rather than approximate. Every value has had the same subtracted, so the total has fallen by exactly ; adding back at the end restores it. **The choice of affects only how easy the arithmetic is**, never the answer — and this method is the ancestor of the step-deviation formula you meet for grouped data in Class 10.
Worked example 1. Find the mean of .
Worked example 2 — with units. The heights of five students are cm, cm, cm, cm and cm. Find the mean height.
The check that costs nothing: a mean must lie between the smallest and the largest value. Here , so the answer is at least possible. A mean outside that range is always an arithmetic slip, usually a missed value in the sum.
Worked example 3 — the assumed-mean shortcut. Find the mean of without adding five three-digit numbers.
Take an assumed mean and work with the deviations :
Check directly: , and , as required.
Why the shortcut is exact rather than approximate. Every value has had the same subtracted, so the total has fallen by exactly ; adding back at the end restores it. **The choice of affects only how easy the arithmetic is**, never the answer — and this method is the ancestor of the step-deviation formula you meet for grouped data in Class 10.
Formula
How do you find the mean when values come with frequencies?
Multiply each value by its frequency, add those products, and divide by the total frequency.
Worked example. Find the mean of this distribution:
- with frequency
- with frequency
- with frequency
- with frequency
- with frequency
The products are , , , and , so
The denominator is the total frequency, not the number of rows. There are five distinct values but twenty observations, and dividing by would give — far outside the data. That single error is the most common one in this section, and the range check catches it immediately, since is nowhere between and .
Why the frequencies matter — look at what they did. If all five values had appeared equally often, the mean would be the plain average of , which is . The frequencies here lean towards the upper values, and the mean moved up to . A frequency distribution's mean is a weighted average, and the weights are the frequencies.
Worked example 2 — a real count. In a survey of households the number of members was recorded as: members in households, in , in , in and in . Find the mean household size.
A mean need not be a possible value. No household contains people, and that is not an error — the mean is a balance point for the whole set, not a member of it. **Reporting it as about four members per household is the sensible reading**, and knowing when to round is part of using statistics rather than just calculating it.
Worked example. Find the mean of this distribution:
- with frequency
- with frequency
- with frequency
- with frequency
- with frequency
The products are , , , and , so
The denominator is the total frequency, not the number of rows. There are five distinct values but twenty observations, and dividing by would give — far outside the data. That single error is the most common one in this section, and the range check catches it immediately, since is nowhere between and .
Why the frequencies matter — look at what they did. If all five values had appeared equally often, the mean would be the plain average of , which is . The frequencies here lean towards the upper values, and the mean moved up to . A frequency distribution's mean is a weighted average, and the weights are the frequencies.
Worked example 2 — a real count. In a survey of households the number of members was recorded as: members in households, in , in , in and in . Find the mean household size.
A mean need not be a possible value. No household contains people, and that is not an error — the mean is a balance point for the whole set, not a member of it. **Reporting it as about four members per household is the sensible reading**, and knowing when to round is part of using statistics rather than just calculating it.
How do you find the median for an odd and an even number of values?
Arrange the data in order first. Then for an odd count the median is the middle value, and for an even count it is the average of the two middle values.
Worked example 1 — odd count. Find the median of .
Already in order, with , so the median is the th value:
Worked example 2 — even count. Find the median of .
Here , so average the th and th values:
Worked example 3 — unsorted data, which is the real test. Find the median of .
Sort first: . With the median is the th value, which is .
**Taking the middle of the unsorted list would have given as well here by luck**, but with the original order the fourth entry is already a different number in most rearrangements. Sorting is not a tidying step — it is the definition, and skipping it is the single biggest source of wrong medians.
Worked example 4 — median of a frequency distribution. Find the median of the distribution from the previous section: with frequencies .
The total frequency is , so the median is the average of the th and th observations. Build the cumulative frequency:
- up to :
- up to :
- up to :
- up to :
- up to :
The th and th observations both fall in the row that reaches , so both are :
Compare the two answers for the same data: the mean was and the median is . The distribution leans slightly to the right, so the mean sits a little above the median — and the gap between mean and median is itself a description of the data's shape.
Worked example 1 — odd count. Find the median of .
Already in order, with , so the median is the th value:
Worked example 2 — even count. Find the median of .
Here , so average the th and th values:
Worked example 3 — unsorted data, which is the real test. Find the median of .
Sort first: . With the median is the th value, which is .
**Taking the middle of the unsorted list would have given as well here by luck**, but with the original order the fourth entry is already a different number in most rearrangements. Sorting is not a tidying step — it is the definition, and skipping it is the single biggest source of wrong medians.
Worked example 4 — median of a frequency distribution. Find the median of the distribution from the previous section: with frequencies .
The total frequency is , so the median is the average of the th and th observations. Build the cumulative frequency:
- up to :
- up to :
- up to :
- up to :
- up to :
The th and th observations both fall in the row that reaches , so both are :
Compare the two answers for the same data: the mean was and the median is . The distribution leans slightly to the right, so the mean sits a little above the median — and the gap between mean and median is itself a description of the data's shape.
How do you find a missing value or correct a mean that used a wrong entry?
Turn the mean back into a total. Every question of this kind is solved by the same first line: .
Worked example 1 — a missing observation. The mean of six numbers is . Five of them are and . Find the sixth.
The five known values add to , so the sixth is
Check: , as required.
Worked example 2 — a misread entry. The mean of observations was calculated as . It was then found that an observation of had been read as . Find the correct mean.
Start from the wrong total, remove the wrong value and put the right one in:
Adjust the total, never the mean. Adding to the mean instead of to the total is the standard error here; the entry rose by , so the mean rises by only , which is exactly what happened.
Worked example 3 — changing every value. The mean of numbers is . Find the new mean if every number is increased by , and if every number is multiplied by .
The total was .
- Increasing each by adds to the total, so the new mean is
- Doubling each doubles the total, so the new mean is
So the mean follows whatever you do to every value — add and it rises by ; multiply by and it doubles. That is worth knowing as a rule, because it saves recomputing a whole table, and in Class 11 it is stated formally as the effect of a change of origin and scale.
Worked example 4 — a combined mean. In a class, boys have a mean mark of and girls a mean of . Find the mean for the whole class.
Go back to totals for each group:
**Notice that the answer is , not .** The plain average of and would be , but there are more boys than girls, so the combined mean sits nearer the boys' value. Never average two means unless the two groups are the same size — add the totals instead.
Worked example 1 — a missing observation. The mean of six numbers is . Five of them are and . Find the sixth.
The five known values add to , so the sixth is
Check: , as required.
Worked example 2 — a misread entry. The mean of observations was calculated as . It was then found that an observation of had been read as . Find the correct mean.
Start from the wrong total, remove the wrong value and put the right one in:
Adjust the total, never the mean. Adding to the mean instead of to the total is the standard error here; the entry rose by , so the mean rises by only , which is exactly what happened.
Worked example 3 — changing every value. The mean of numbers is . Find the new mean if every number is increased by , and if every number is multiplied by .
The total was .
- Increasing each by adds to the total, so the new mean is
- Doubling each doubles the total, so the new mean is
So the mean follows whatever you do to every value — add and it rises by ; multiply by and it doubles. That is worth knowing as a rule, because it saves recomputing a whole table, and in Class 11 it is stated formally as the effect of a change of origin and scale.
Worked example 4 — a combined mean. In a class, boys have a mean mark of and girls a mean of . Find the mean for the whole class.
Go back to totals for each group:
**Notice that the answer is , not .** The plain average of and would be , but there are more boys than girls, so the combined mean sits nearer the boys' value. Never average two means unless the two groups are the same size — add the totals instead.
Exam tip
What layout keeps a mean or median question safe?
**Write , write the total, then divide — as three separate steps. Most of the marks in this chapter are for the structure, and a single displayed division is hard to award.
- State or explicitly. For a frequency table, write the total frequency under the column; it is the denominator and the commonest thing to get wrong
- Show the column** as a list of products before adding. An unexplained earns less than a visible
- Sort the data and write the sorted list before finding a median. The sorted list is itself a creditable step
- Quote the position formula and then the value: *, so the median is the average of the th and th values*
- **For missing or corrected values, start with . Write that line first every time; the rest is arithmetic
- Adjust the total, not the mean, when an entry is replaced
- Check the answer lies inside the data range, and carry units into the final answer
The distinction to keep clear. The mean** can be a value that no observation takes — members in a household — while the median of an odd-sized data set is always one of the actual values. For an even count the median can also fall between two observations. Neither result is an error, and a question asking which average better describes this data is asking you to notice the outlier, not to recalculate.
- State or explicitly. For a frequency table, write the total frequency under the column; it is the denominator and the commonest thing to get wrong
- Show the column** as a list of products before adding. An unexplained earns less than a visible
- Sort the data and write the sorted list before finding a median. The sorted list is itself a creditable step
- Quote the position formula and then the value: *, so the median is the average of the th and th values*
- **For missing or corrected values, start with . Write that line first every time; the rest is arithmetic
- Adjust the total, not the mean, when an entry is replaced
- Check the answer lies inside the data range, and carry units into the final answer
The distinction to keep clear. The mean** can be a value that no observation takes — members in a household — while the median of an odd-sized data set is always one of the actual values. For an even count the median can also fall between two observations. Neither result is an error, and a question asking which average better describes this data is asking you to notice the outlier, not to recalculate.
Did you know
Why do the deviations from a mean always add up to zero?
Take the first worked example, , whose mean is . Now subtract the mean from each value:
Add them: the total is exactly . Try it with any data at all and the same thing happens.
And it must. The deviations add to
and since , the second term is , so the whole thing is zero. The mean is precisely the number that makes the shortfalls and the excesses cancel.
That gives the mean a physical meaning. Imagine the values marked on a metre rule and equal weights hung at each mark. The rule balances exactly at the mean, because balancing is the condition that the turning effects on the two sides — the deviations — cancel. In Physics the same calculation is the centre of mass, and the formula is your with masses in place of frequencies.
It also explains the lane of shopkeepers. One value of is a long way from the rest, so it exerts a large turning effect, and the balance point has to shift far up the rule to compensate. The median, which only counts how many values sit on each side rather than how far away they are, does not move at all.
And it gives you a free check on any mean. Compute the deviations and add them: if they do not cancel, the mean is wrong. For the household survey above, with mean , the weighted deviations are , as required.
One thing the cancellation does not tell you is how spread out the data is, since the deviations always sum to zero however scattered the values are. Fixing that needs the squares of the deviations, and that is where standard deviation comes from in Class 11.
Add them: the total is exactly . Try it with any data at all and the same thing happens.
And it must. The deviations add to
and since , the second term is , so the whole thing is zero. The mean is precisely the number that makes the shortfalls and the excesses cancel.
That gives the mean a physical meaning. Imagine the values marked on a metre rule and equal weights hung at each mark. The rule balances exactly at the mean, because balancing is the condition that the turning effects on the two sides — the deviations — cancel. In Physics the same calculation is the centre of mass, and the formula is your with masses in place of frequencies.
It also explains the lane of shopkeepers. One value of is a long way from the rest, so it exerts a large turning effect, and the balance point has to shift far up the rule to compensate. The median, which only counts how many values sit on each side rather than how far away they are, does not move at all.
And it gives you a free check on any mean. Compute the deviations and add them: if they do not cancel, the mean is wrong. For the household survey above, with mean , the weighted deviations are , as required.
One thing the cancellation does not tell you is how spread out the data is, since the deviations always sum to zero however scattered the values are. Fixing that needs the squares of the deviations, and that is where standard deviation comes from in Class 11.
Exam relevance
How are mean and median tested in JEE-level statistics?
This is foundation work whose formulas are reused almost unchanged in Class 11, where statistics is a JEE Main topic.
Where it leads. The Class 10 chapter extends to grouped data using class marks and the step-deviation method — the assumed-mean shortcut you used above. In Class 11, variance and standard deviation are built directly on the mean, and the shortcut formula is derived using exactly the fact that . Knowing why the deviations cancel makes that derivation readable rather than magical.
Where the specific results reappear. The combined mean of two groups is set directly in JEE Main, often with the combined variance as well. The effect of change of origin and scale — add a constant and the mean shifts, multiply and it scales — is a standard assertion-reason item, with the twist that variance is unaffected by adding a constant but multiplied by the square when scaling. And in Physics, for the centre of mass is the same calculation you have been doing.
Question types to expect. At this level: direct means, medians, missing observations and corrected means. In competitive papers: mean and variance from a frequency table, combined statistics for two groups, and the effect of altering every observation. Numericals dominate; the theory appears as assertion-reason.
The single trap that costs marks. Dividing by the number of distinct values instead of . In a Class 11 variance question this error is fatal and invisible — every subsequent line looks correct. **Write under the frequency column and use that number, and run the range check on the mean before going further.
A second trap worth naming. Averaging two means when the groups differ in size. It is the most frequently set distractor in combined-mean questions, and the correct route is always through totals.
Board versus competitive emphasis. ICSE marks the columns, the totals and the stated formula; a competitive paper marks one number. The habit that carries across is turning a mean back into a total** — is the first line of almost every statistics question you will meet for the next three years.
Where it leads. The Class 10 chapter extends to grouped data using class marks and the step-deviation method — the assumed-mean shortcut you used above. In Class 11, variance and standard deviation are built directly on the mean, and the shortcut formula is derived using exactly the fact that . Knowing why the deviations cancel makes that derivation readable rather than magical.
Where the specific results reappear. The combined mean of two groups is set directly in JEE Main, often with the combined variance as well. The effect of change of origin and scale — add a constant and the mean shifts, multiply and it scales — is a standard assertion-reason item, with the twist that variance is unaffected by adding a constant but multiplied by the square when scaling. And in Physics, for the centre of mass is the same calculation you have been doing.
Question types to expect. At this level: direct means, medians, missing observations and corrected means. In competitive papers: mean and variance from a frequency table, combined statistics for two groups, and the effect of altering every observation. Numericals dominate; the theory appears as assertion-reason.
The single trap that costs marks. Dividing by the number of distinct values instead of . In a Class 11 variance question this error is fatal and invisible — every subsequent line looks correct. **Write under the frequency column and use that number, and run the range check on the mean before going further.
A second trap worth naming. Averaging two means when the groups differ in size. It is the most frequently set distractor in combined-mean questions, and the correct route is always through totals.
Board versus competitive emphasis. ICSE marks the columns, the totals and the stated formula; a competitive paper marks one number. The habit that carries across is turning a mean back into a total** — is the first line of almost every statistics question you will meet for the next three years.
Key takeaways
What should you be able to do with averages before moving on?
Two averages, each with its own strengths, and one identity that connects them to everything later.
- Mean of raw data: , and it must lie between the smallest and largest value
- Mean of a frequency distribution: — the denominator is the total frequency, not the number of rows
- Assumed-mean shortcut: , exact for any choice of
- Median: sort first, then the th value for odd , or the average of the two middle values for even
- From a frequency table, use cumulative frequency to locate the middle observation
- ** is the first line for every missing-value and corrected-mean question — adjust the total, never the mean
- Add a constant to every value and the mean shifts by it; multiply and it scales
- Combine two groups through totals, never by averaging the two means
- Deviations from the mean always sum to zero, which makes the mean a balance point and gives you a free check
- The mean is pulled by extreme values; the median is not**
The test of this chapter is the corrected-mean question. The mean of ten observations is and an entry of was read as — write down the correct mean in two lines, and check that the mean moved by rather than by .
- Mean of raw data: , and it must lie between the smallest and largest value
- Mean of a frequency distribution: — the denominator is the total frequency, not the number of rows
- Assumed-mean shortcut: , exact for any choice of
- Median: sort first, then the th value for odd , or the average of the two middle values for even
- From a frequency table, use cumulative frequency to locate the middle observation
- ** is the first line for every missing-value and corrected-mean question — adjust the total, never the mean
- Add a constant to every value and the mean shifts by it; multiply and it scales
- Combine two groups through totals, never by averaging the two means
- Deviations from the mean always sum to zero, which makes the mean a balance point and gives you a free check
- The mean is pulled by extreme values; the median is not**
The test of this chapter is the corrected-mean question. The mean of ten observations is and an entry of was read as — write down the correct mean in two lines, and check that the mean moved by rather than by .