A cumulative frequency curve. The x-value where it reaches \tfrac{n}{2} is the median.

Statistics: grouped data

Q: What is a class mark?

The middle of a class interval: \dfrac{\text{lower limit} + \text{upper limit}}{2}.

Q: What is the empirical relationship between mean, median and mode?

For moderately skewed data, 3\,\text{median} = \text{mode} + 2\,\text{mean}.

When data is grouped into class intervals (0–10, 10–20, …) the individual values are lost, so the mean, median and mode are estimated with formulas that use the class marks, cumulative frequencies and the modal class.

Formulas

Mean (x = class mark, the middle of each interval)

\overset{x}{ˉ} = \frac{\sum f x}{\sum f}

Median (l = lower limit of the median class, cf = cumulative frequency before it, h = class width)

median = l + \frac{\frac{n}{2} - c f}{f} \times h

Mode (f₁ = modal class frequency, f₀ and f₂ the classes either side)

mode = l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h

Worked examples

Example 1. Find the mean of: 0–10 (3), 10–20 (5), 20–30 (8), 30–40 (4).

Class marks 5, 15, 25, 35. $f x = 15$ , 75, 200, 140, total 430.
$\sum f = 20, so \overset{x}{ˉ} = \frac{430}{20} = 21.5$ .

Answer: 21.5

Example 2. Find the median of the same data.

$n = 20, \frac{n}{2} = 10$ . Cumulative frequencies: 3, 8, 16, 20, so the median class is 20–30.
$l = 20, c f = 8, f = 8, h = 10$ : median $= 20 + \frac{10 - 8}{8} \times 10 = 22.5$ .

Answer: 22.5

Example 3. Find the mode of the same data.

The modal class is $20 - 30 (f_{1} = 8)$ , with $f_{0} = 5$ and $f_{2} = 4$ .
Mode $= 20 + \frac{8 - 5}{16 - 5 - 4} \times 10 = 20 + \frac{30}{7} \approx 24.29$ .

Answer: about 24.29

Example 4. A distribution has median 30 and mean 28. Estimate the mode.

Use the empirical relation $mode = 3 median - 2 mean$ .
$3 \times 30 - 2 \times 28 = 90 - 56 = 34$ .

Answer: 34

Common mistakes

Using the class limits instead of the class mark for the mean. The class mark of 10–20 is 15.
Taking the class with the largest cumulative frequency as the median class. It is the first class whose cumulative frequency reaches $\frac{n}{2}$ .
Using the cumulative frequency OF the median class for $c f$ . It must be the one just before it.

Practice questions with answers

1. For grouped data: class 0–10 (f=4), 10–20 (f=6), 20–30 (f=10). Find mean (direct method).

Show answer

Answer: B) 18

Marks 5, 15, 25. Σfx = 20+90+250 = 360. Σf = 20. Mean = 18.

2. In a grouped frequency table, what does the symbol 'fi' represent?

AThe class mark of a class
BThe frequency of a class
CThe total of all frequencies
DThe class width

Show answer

Answer: B) The frequency of a class

fi denotes the frequency, i.e. the number of observations falling in the i-th class.

3. Which formula is used to find the median of grouped data?

A $l + \frac{n /2 - c f}{f} \times h$
B $l + \frac{f - n /2}{c f} \times h$
C $a + \frac{\sum f \cdot d}{\sum f}$
D $l + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h$

Show answer

Answer: A)

l + \frac{n /2 - c f}{f} \times h

The median of grouped data is given by Median =

l + \frac{n /2 - c f}{f} \times h

, where

l

is the lower boundary of the median class,

c f

is the cumulative frequency before it,

f

is its frequency and

h

is the class size.

4. The class with the highest frequency in a frequency distribution is called the:

AModal class
BMedian class
CMean class
DMid class

Show answer

Answer: A) Modal class

The class interval having the maximum frequency is called the modal class. The mode lies in this class.

5. What does 'cumulative frequency' of a class mean?

AThe frequency of that class only
BThe sum of frequencies of that class and all classes before it
CThe total of all frequencies in the table
DThe midpoint of the class

Show answer

Answer: B) The sum of frequencies of that class and all classes before it

Cumulative frequency (less than type) is obtained by adding up the frequency of a class and all classes preceding it.

6. Find the mean of: 4, 6, 8, 10, 12.

Show answer

Answer: A) 8

Sum = 40,

n = 5

. Mean =

\frac{40}{5} = 8

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Frequently asked questions

What is a class mark?

The middle of a class interval: $\frac{lower limit + upper limit}{2}$ .

What is the modal class?

The class interval with the highest frequency.

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

For moderately skewed data, $3 median = mode + 2 mean$ .

What is an ogive?

A cumulative frequency curve. The $x$ -value where it reaches $\frac{n}{2}$ is the median.

Statistics: grouped data

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

Related topics