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Here is a job advertisement: “Our team of nine earns an average of ₹11.5 lakh a year.” It sounds great until you look at the payslips. Eight people earn between ₹4 and ₹7 lakh. The ninth, the founder, takes ₹60 lakh. The ad is technically true, and still misleading. The trick is that “average” can mean three different things, and they can disagree wildly.
The three averages in one minute
All three try to answer the same question, “what is a typical value?”, in different ways. Take our nine salaries in lakh rupees per year: 4, 4.5, 5, 5, 5.5, 6, 6.5, 7 and 60.
Eight bars are close to each other. The ninth is almost ten times taller. The mean has to reflect all nine, so it is dragged toward the tall one.
The mean is more than double the median. A new hire expecting ₹11.5 lakh would be disappointed.
How to find each one
Mean
Add up all the values and divide by how many there are. Here 4 + 4.5 + 5 + 5 + 5.5 + 6 + 6.5 + 7 + 60 = 103.5, and 103.5 ÷ 9 = 11.5. The mean uses every number, which is its strength (it is fair to all data) and its weakness (it is easily dragged around by one extreme number).
Median
Sort the values and pick the middle one. With nine values the middle is the fifth, which is 5.5. If the count is even, there are two middle values, and you average them. For 3, 5, 7, 9 the median is (5 + 7) ÷ 2 = 6. The median ignores how large the extreme values are. It only cares about their position.
Mode
The value that appears most often. In our data 5 appears twice and everything else once, so the mode is 5. A set can have no mode (all values different) or several. The mode is the only one of the three that works for categories: the most common shirt size, the most popular bus route.
- 1Sort the numbers4, 4.5, 5, 5, 5.5, 6, 6.5, 7, 60.
- 2Count themThere are 9 values. The middle position is (9 + 1) ÷ 2 = 5.
- 3Read the 5th valueIt is 5.5. If the count were even, average the two middle ones.
How one number bends the mean
Let us do an experiment. Keep the other eight salaries fixed and change only the ninth. As the top salary grows, the mean rises steadily. The median does not move at all, because the top person is still the top person and the middle is still the fifth value.
With a top salary of 10 the two are close (5.9 and 5.5). At 100, the mean is 15.9 and the median is still 5.5. This is why the median is called a robust measure.
The gap between the two (6 lakh) is a quick signal that the data is lopsided. When the mean is far above the median, a few big values are at work.
Skew: when the three averages disagree
In a perfectly symmetric bell curve all three averages sit at the same point. In a right-skewed distribution, which has a long tail of big values, the order is usually mode, then median, then mean, from left to right. Incomes, house prices, hospital stays and the time to fix a bug all look like this. Few values are huge, and many are small to medium.
Drawn from a standard gamma curve: mode 1, median about 1.68, mean 2. The long right tail pulls the mean furthest. If the mean is above the median, suspect a right skew.
- True, and technically correct
- Raised by the founder’s ₹60 lakh
- Describes nobody in the room
- Half the team earns more, half less
- Not moved by the founder’s pay
- Matches what a new hire will likely see
Whenever you read a headline with “average”, ask: mean or median?
Which one should you use?
Good reports give two numbers, such as “median ₹5.5 lakh, mean ₹11.5 lakh”. The gap itself is information.
Other kinds of average you will meet
The weighted mean
Sometimes values do not count equally. A course gives 20% to assignments, 30% to a midterm and 50% to the final. You scored 80, 60 and 70. The plain mean is (80 + 60 + 70) ÷ 3 = 70. The weighted mean is 0.2 × 80 + 0.3 × 60 + 0.5 × 70 = 16 + 18 + 35 = 69. The final counts the most, so it pulls harder.
The weights decide the result. A grade calculator or the statistics tool above does this arithmetic for you.
The trimmed mean
Drop the highest and lowest values before averaging. Judges in diving and gymnastics do this to blunt a biased score. For our nine salaries, drop 4 and 60 and the mean of the remaining seven is 39.5 ÷ 7 ≈ 5.6 lakh, very close to the median.
The geometric mean
For growth rates you multiply rather than add, so the right average is the geometric mean. A fund that gains 50% one year and loses 50% the next has an arithmetic mean of 0%, but your money is down: 1.5 × 0.5 = 0.75. The true yearly rate is √0.75 − 1 ≈ −13.4%. The articles on up 50 percent, down 50 percent and average return versus CAGR walk through this trap.
Five ways averages mislead
- The outlier hides in the mean. One huge value shifts it. Always check the median.
- Two groups hiding in one. If data has two clusters (say, junior and senior staff), no single average describes either group. Split it first.
- Averages of averages. The mean of two city averages is not the national average unless the cities are the same size. Use a weighted mean.
- Percentages are not added. Average the underlying counts, not the percentages.
- The mean of nothing. The average family has 2.4 children, but no family does. The mean need not be a possible value, which is fine for planning and silly for describing.
Quick practice
Five friends spend ₹200, ₹250, ₹300, ₹300 and ₹1,950 on a trip dinner. The mean is 3,000 ÷ 5 = ₹600, the median ₹300, and the mode ₹300. Which should the restaurant quote as the typical bill? The median, because one person ordered an expensive meal. Which should the group use to split the total fairly? Neither: the total of ₹3,000 is what matters, so the mean times five is the figure to split. The same data, two different questions, two different averages.
Questions people ask
What is the difference between mean, median and mode?
The mean is the sum divided by the count, the median is the middle value of the sorted list, and the mode is the most frequent value.
Why is the median better for salaries?
A few very high salaries pull the mean up but do not change the middle value, so the median shows what a typical person earns.
Can the mean be higher than every value except one?
Yes. With eight values near 5 and one value of 60, the mean is 11.5, which is above eight of the nine values.
What if there are two middle values?
When the count is even, the median is the average of the two middle values. For 3, 5, 7, 9 it is 6.
Can a data set have more than one mode?
Yes. If two values tie for the most frequent, the data is bimodal. If no value repeats, it has no mode.
What is a weighted average?
An average in which some values count more than others, found by multiplying each value by its weight and adding. Grades often use it.