Class 10 · Chapter 13

Statistics

Measures of central tendency for grouped data: mean, median, mode and their empirical relationship.

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Detailed theory

For a frequency distribution with classes and midpoints

x_i

and frequencies

f_i

, direct method:

\bar x = \dfrac{\sum f_i x_i}{\sum f_i}

. Assumed-mean method: pick

a

near the middle;

d_i = x_i - a

; then

\bar x = a + \dfrac{\sum f_i d_i}{\sum f_i}

. Step-deviation: with

u_i = (x_i - a)/h

\bar x = a + h\cdot\dfrac{\sum f_i u_i}{\sum f_i}

Key concepts & formulas

Mean of grouped data:

\bar x = \dfrac{\sum f_i x_i}{\sum f_i}

Median

= l + \dfrac{n/2 - cf}{f}\cdot h

Mode

= l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\cdot h

Empirical relation:

3\text{Median} = 2\text{Mean} + \text{Mode}

Cumulative frequency

=

running total of frequencies.

Assumed mean method:

\bar x = a + \dfrac{\sum f_i d_i}{\sum f_i}

where

d_i = x_i - a