Median Formula, Mean Formula and Mode Formula Explained

What does mean in math, and how median and mode differ

Students often ask what does mean in math stand for. The mean is the everyday average: add all the values and share the total equally. The median is the middle value once the data is sorted. The mode is the value that appears most often. All three are called measures of central tendency, because each tries to describe the whole data with a single typical number.

Take the incomes 12, 14, 15, 16 and 93 thousand rupees. The mean is 30 thousand, which describes nobody in the list. The median, 15 thousand, describes the group far better.

Median formula for ungrouped data (odd and even n)

First arrange the values in ascending order. Skipping this step is the single biggest reason for wrong medians.

Mean formula for lists and frequency tables

The formula of mean for a plain list is $\bar{x} = \tfrac{\sum x_i}{n}$. The mean of 3, 8, 10, 15 is $\tfrac{36}{4} = 9$. This is also the arithmetic mean formula, and the average formula in maths that you use for marks, runs or temperatures.

For a frequency table, the class 10 Statistics chapter teaches three methods. Each class is replaced by its class mark $x_i$ (the mid-point).

1. Direct method: $\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}$.
2. Assumed mean method: choose a class mark $a$ near the centre, let $d_i = x_i - a$, then $\bar{x} = a + \dfrac{\sum f_i d_i}{\sum f_i}$.
3. Step deviation method: divide the deviations by the class size $h$, $u_i = \dfrac{x_i - a}{h}$, then $\bar{x} = a + h \cdot \dfrac{\sum f_i u_i}{\sum f_i}$.

The three methods always give the same mean. The second and third only shrink the numbers so the multiplication is quicker. The same $u_i$ column is reused for variance on the standard deviation formula page.

One frequency table solved for mean, median and mode

Daily travel time (minutes) of 40 office workers:

Median formula for grouped data, term by term

This is the median formula class 10 students use in board exams. Every letter refers to the median class, the class whose cumulative frequency first reaches or crosses $\tfrac{n}{2}$.

• $l$: lower limit of the median class.
• $n$: total frequency, $\sum f_i$. Use $\tfrac{n}{2}$ here, not $\tfrac{n+1}{2}$, whether n is odd or even.
• $cf$: cumulative frequency of the class just before the median class.
• $f$: frequency of the median class itself.
• $h$: class size (width).

The idea is linear interpolation. You know $cf$ values lie below $l$, and you need $\tfrac{n}{2} - cf$ more of the $f$ values spread evenly across a width of $h$.

Mode formula and the empirical relation

For ungrouped data, the mode is simply the most frequent value. In 3, 5, 7, 5, 9, 7, 5, 2 the value 5 appears three times, so the mode is 5. A list can have two modes (bimodal) or none.

The formula of mode for grouped data, $l + \tfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$, is the mode formula class 10 uses. Here $f_1$ is the frequency of the modal class and $f_0, f_2$ are the frequencies of its neighbours. If the modal class is the first class, take $f_0 = 0$.

For a moderately skewed distribution, the three averages are linked by Karl Pearson's empirical relation:

From the table above: $3(36) - 2(36.25) = 108 - 72.5 = 35.5$, close to the 35.38 found directly. Use it when a question gives two of the three and asks for the third. For a perfectly symmetric distribution, mean, median and mode are equal.

Arithmetic, geometric and harmonic mean formulas

For positive numbers $x_1, x_2, \ldots, x_n$:

Two facts worth memorising for sequences and series questions: $\text{AM} \ge \text{GM} \ge \text{HM}$, with equality only when all numbers are equal, and for two numbers $\text{GM}^2 = \text{AM} \times \text{HM}$. Check: $6.5 \times \tfrac{72}{13} = 36 = 6^2$.

The geometric mean of two numbers is also called the mean proportional. The mean proportional formula between a and b is $\sqrt{ab}$, so between 4 and 9 it is 6, because $4 : 6 = 6 : 9$. For three numbers, the GM of 2, 8, 32 is $\sqrt[3]{512} = 8$. Use the harmonic mean for average speed over equal distances: 40 km/h one way and 60 km/h back gives $\tfrac{2 \times 40 \times 60}{100} = 48$ km/h, not 50.

Weighted average formula

When some values count more than others, multiply each by its weight:

A student scores 78 in a test worth 3 credits, 85 in one worth 4 credits and 62 in one worth 2 credits. Weighted average $= \dfrac{78(3) + 85(4) + 62(2)}{9} = \dfrac{698}{9} \approx 77.56$. The plain average of the three marks would be 75, which ignores the credits. The grouped mean $\tfrac{\sum f_i x_i}{\sum f_i}$ is a weighted average where the frequencies are the weights. For marks written as a share of the total, the percentage formula page shows the conversions.

Exam checklist for statistics questions

• Write the formula first, then a line listing $l, n, cf, f, h$ with their values. Examiners often give step marks for this.
• Draw the cumulative frequency column before naming the median class.
• Mode uses the class with the highest frequency; median uses the class where cf crosses $\tfrac{n}{2}$. They are often different classes.
• In assumed mean questions, any $a$ works, but a middle class mark keeps numbers small.
• Round only the final answer, usually to two decimal places.

Measures of spread (range, mean deviation, variance, SD) follow in class 11. The complete formula list gives you one sheet of revision across chapters.