Mean, Median and Mode: Central Tendency | NIMCET

Quick answer: For data $x_1, \ldots, x_n$ with frequencies $f_i$ , the mean is $\bar{x} = \frac{\sum f_ix_i}{\sum f_i}$ . The median is the middle value of the ordered data; for grouped data, $\text{Median} = l + \frac{\frac{N}{2} - c}{f} \times h$ . The mode is the most frequent value; for grouped data, $\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$ . For moderately skewed data, $\text{Mode} \approx 3\,\text{Median} - 2\,\text{Mean}$ .

Frequency distributions

Type	Description
Raw data	Individual observations
Discrete frequency distribution	Values with their frequencies
Grouped (continuous) distribution	Class intervals with frequencies
Cumulative frequency (cf)	Running total of frequencies

Mean

Data	Formula
Raw	$\bar{x} = \frac{\sum x_i}{n}$
Frequency	$\bar{x} = \frac{\sum f_ix_i}{\sum f_i}$
Assumed mean A, step h	$\bar{x} = A + h\frac{\sum f_iu_i}{\sum f_i}$ , with $u_i = \frac{x_i - A}{h}$
Combined mean	$\bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$

Grouped data uses class midpoints for $x_i$ .

Properties: the sum of deviations from the mean is 0. Adding k to every value adds k to the mean; multiplying every value by k multiplies the mean by k.

Median

Data	Rule
Raw, n odd	The $\left(\frac{n + 1}{2}\right)$ th value
Raw, n even	Average of the $\frac{n}{2}$ th and $\left(\frac{n}{2} + 1\right)$ th values
Grouped	$l + \frac{\frac{N}{2} - c}{f} \times h$

For grouped data: l is the lower limit of the median class, N the total frequency, c the cumulative frequency before the median class, f its frequency and h the class width.

Mode

For grouped data, with modal class (highest frequency) having lower limit l, frequency $f_1$ , previous frequency $f_0$ , next frequency $f_2$ and width h:

$\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$

Empirical relation: $\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$ (approximately, for moderately skewed data).

Effect of changes in data

Change to each value	Mean, median, mode
Add k	All increase by k
Multiply by k	All are multiplied by k

Worked examples

Example 1: raw data

Find the mean and median of 7, 3, 9, 5, 11, 3.

Step 1: Mean $= \frac{38}{6} = \frac{19}{3}$ .

Step 2: Ordered: 3, 3, 5, 7, 9, 11. Median $= \frac{5 + 7}{2} = 6$ . Mode $= 3$ .

Example 2: discrete frequency

x	1	2	3	4
f	3	5	8	4

Mean $= \frac{3 + 10 + 24 + 16}{20} = \frac{53}{20} = 2.65$ .

Example 3: grouped median

Class	0–10	10–20	20–30	30–40	40–50
f	5	8	12	10	5
cf	5	13	25	35	40

Step 1: $\frac{N}{2} = 20$ ; the median class is 20–30.

Step 2:

$\text{Median} = 20 + \frac{20 - 13}{12} \times 10 = 20 + 5.83 = 25.83$

Example 4: grouped mode

For the same table, the modal class is 20–30 with $f_1 = 12$ , $f_0 = 8$ , $f_2 = 10$ :

$\text{Mode} = 20 + \frac{12 - 8}{24 - 8 - 10} \times 10 = 20 + \frac{40}{6} \approx 26.67$

Example 5: combined mean

Class A (30 students) averages 60 and class B (20 students) averages 70. Find the overall mean.

$\frac{30 \cdot 60 + 20 \cdot 70}{50} = \frac{3200}{50} = 64$

Summary table

Measure	Key formula
Mean	$\frac{\sum f_ix_i}{\sum f_i}$
Grouped median	$l + \frac{\frac{N}{2} - c}{f}h$
Grouped mode	$l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2}h$
Combined mean	$\frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$
Empirical	Mode = 3 Median − 2 Mean

Exam tips

Always sort raw data before finding the median.
Build the cumulative frequency column first for grouped medians.
For "a wrong value was recorded" questions, correct the total, then divide again.

Next, study variance, standard deviation and moments, or see the NIMCET Probability and Statistics chapter.

Common mistakes

Finding the median without arranging the data in order.
Using class limits instead of midpoints for the grouped mean.
Taking c as the cumulative frequency of the median class itself.
Averaging two group means without weighting by group size.

Frequently Asked Questions

What is the formula for the median of grouped data?

Median = l + ((N/2 − c)/f) × h.

What is the empirical relation between mean, median and mode?

Mode ≈ 3 Median − 2 Mean, for moderately skewed data.

How do you find a combined mean?

Weight each group mean by its size: (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂).

What happens to the mean if each value is increased by 5?

The mean also increases by 5.

What is the mode of grouped data?

Mode = l + ((f₁ − f₀)/(2f₁ − f₀ − f₂)) × h, where the modal class has the highest frequency.