Arithmetic mean - ungrouped frequency distribution — lesson. Mathematics State Board, Class 9.

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The mean of the ungrouped frequency distribution can be determined using the formula:

$\overline X = \frac{f_1 x_1 + f_2 x_2 + ... + f_n x_n}{f_1 + f_2 + ... + f_n}$

$= \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i}$

$= \frac{\sum fx}{\sum f}$

Example:

The marks scored by

$20$ students in mathematics are

$100$ ,

$65$ ,

$89$ ,

$55$ ,

$40$ ,

$96$ ,

$65$ ,

$35$ ,

$100$ ,

$55$ ,

$65$ ,

$96$ ,

$89$ ,

$77$ ,

$63$ ,

$77$ ,

$30$ .

Solution:

In the previous sessions, we have learnt how to find the average using the arithmetic mean method and assumed mean method.

Now, we shall learn the

$3^{rd}$ method, which is an ungrouped frequency distribution.

Here, in the data, we can find

$100$ has occured

$4$ times (i.e. frequency is

$4$ ),

$65$ has frequency of

$3$ and so on. Then, the frequency distribution table looks like this:

Marks $x$	Number of students $f$
$30$	$1$
$35$	$2$
$40$	$1$
$55$	$2$
$63$	$1$
$65$	$3$
$77$	$2$
$89$	$2$
$96$	$2$
$100$	$4$

We know the formula to find the mean of ungrouped frequency distribution is

$\overline X = \frac{\sum fx}{\sum f}$

To find the value of

$fx$ , multiply the value of

$x$ and

$f$ of each entry.

Consider for the mark

$30$ . That is,

$30 \times 1 = 30$

Similarly, for the mark

$35$ , we have

$35 \times 2 = 70$ and so on.

Tabulating these values, we get:

Marks $x$	Frequency $f$	$fx$
$30$	$1$	$30$
$35$	$2$	$70$
$40$	$1$	$40$
$55$	$2$	$110$
$63$	$1$	$63$
$65$	$3$	$195$
$77$	$2$	$154$
$89$	$2$	$178$
$96$	$2$	$192$
$100$	$4$	$400$
Total	$\sum f = 20$	$\sum fx = 1432$

Substituting the known values in the above formula, we get:

Mean

$\overline X = \frac{1432}{20}$

$= 71.6$

Therefore, the mean of the given data is

$71.6$ .

We can also find the mean of the ungrouped frequency distribution using the assumed mean method.

Example:

Consider the above example. Let the assumed mean be

$A = 65$ .

Then the new frequency distribution table is given by:

Marks $x$	Deviation $d = x - A$	Frequency $f$	$fd$
$30$	$30 - 65 = - 35$	$1$	$-35$
$35$	$35 - 65 = - 30$	$2$	$-60$
$40$	$40 - 65 = - 25$	$1$	$-25$
$55$	$55 - 65 = - 10$	$2$	$-20$
$63$	$63 - 65 = -2$	$1$	$-2$
$65$	$65 - 65 = 0$	$3$	$0$
$77$	$77 - 65 = 12$	$2$	$24$
$89$	$89 - 65 = 24$	$2$	$48$
$96$	$96 - 65 = 31$	$2$	$62$
$100$	$100 - 65 = 35$	$4$	$140$
Total		$\sum f = 20$	$\sum fd = 132$

Arithmetic mean

$=$ Assumed mean

$+$ Average of sum of deviations

Arithmetic mean

$= 65 + \frac{132}{20}$

$= 65 + 6.6$

$= 71.6$

Therefore, the mean is

$71.6$ .

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6. Arithmetic mean - ungrouped frequency distribution

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