Mathematics: Class 10 — Chapter 13: Statistics

13.1 & 13.2 Mean of Grouped Data

The mean (average) of observations is the sum of all values divided by the total number of observations.

Class Mark

To find the mean of grouped data, each class interval is represented by its mid-point, called the class mark:

Class Mark = (Upper Class Limit + Lower Class Limit) / 2

Methods to Calculate Mean

Direct Method:
Formula: x̄ = Σ(f_i * x_i) / Σf_i
Assumed Mean Method:
Used to simplify calculations when numerical values are large.
Formula: x̄ = a + Σ(f_i * d_i) / Σf_i
Where a is the assumed mean, and d_i = x_i - a (deviation of a from each x_i).
Step-deviation Method:
Further simplifies calculations when deviations have a common factor.
Formula: x̄ = a + [Σ(f_i * u_i) / Σf_i] * h
Where h is the class size, and u_i = (x_i - a) / h.

13.3 Mode of Grouped Data

The mode is the value that occurs most frequently in a dataset.

Modal Class & Formula

In grouped data, the class with the maximum frequency is called the modal class. The mode lies within this class and is calculated as:

Mode = l + [ (f_1 - f_0) / (2f_1 - f_0 - f_2) ] * h

l: Lower limit of the modal class
h: Size of the class interval
f_1: Frequency of the modal class
f_0: Frequency of the class preceding the modal class
f_2: Frequency of the class succeeding the modal class

13.4 Median of Grouped Data

The median is a measure of central tendency that gives the value of the middle-most observation.

Median Class & Formula

To find the median, construct a cumulative frequency (cf) table. Locate the median class whose cumulative frequency is greater than and nearest to n/2 (where n is the total frequency).

Median = l + [ (n/2 - cf) / f ] * h

l: Lower limit of the median class
n: Number of observations (Σf_i)
cf: Cumulative frequency of the class preceding the median class
f: Frequency of the median class
h: Class size

Empirical Relationship

There is an empirical relationship between the three measures of central tendency: