CBSE Class 10 Mathematics Statistics Notes Set 08

Measures of Central Tendency

A single number that represents the whole data is called a measure of central tendency. At this level there are three: the mean (average), the median (middle value) and the mode (most frequent value).
For ungrouped data with n values: mean = sum of values divided by n; the median is the middle value of the data arranged in order (average of the two middle values if n is even); the mode is the value that occurs most often.

Mean of Grouped Data

For a frequency distribution with class marks \( x_i \) and frequencies \( f_i \), the class mark of a class is half the sum of its lower and upper limits.
Direct method: \( \bar{x} = \dfrac{\sum f_i x_i}{\sum f_i} \)
Assumed mean method: choose a convenient class mark a and let \( d_i = x_i - a \). Then \( \bar{x} = a + \dfrac{\sum f_i d_i}{\sum f_i} \).
Step-deviation method: when all the \( d_i \) have a common factor h (the class width), let \( u_i = \dfrac{x_i - a}{h} \). Then \( \bar{x} = a + h \dfrac{\sum f_i u_i}{\sum f_i} \).
All three methods give the same answer; the shortcut methods only reduce the arithmetic.

Mode of Grouped Data

The class with the highest frequency is the modal class. If l is its lower limit, h its width, f1 its frequency, f0 the frequency of the class before it and f2 the frequency of the class after it, then
\( \text{Mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h \)
When classes are written as 10–19, 20–29 (inclusive form), first convert them to true boundaries 9.5–19.5, 19.5–29.5 before using the formula.

Median of Grouped Data

Find n = sum of frequencies and the cumulative frequencies. The median class is the first class whose cumulative frequency is at least n/2. If l is its lower limit, cf the cumulative frequency of the class before it, f its frequency and h its width, then
\( \text{Median} = l + \dfrac{\frac{n}{2} - cf}{f} \times h \)

Empirical Relation and Ogives

For moderately skewed data, \( 3\,\text{Median} = \text{Mode} + 2\,\text{Mean} \), so any one measure can be estimated from the other two.
A less-than ogive is drawn by plotting the upper class limits against the less-than cumulative frequencies; a more-than ogive uses the lower limits and the more-than cumulative frequencies. The x-coordinate of the point where the two ogives cross is the median. Also, the median can be read from a less-than ogive by locating n/2 on the vertical axis.

Worked Example 1: Mean

Find the mean of the following data.
Class: 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 5, 8, 12, 10, 5.
Class marks are 5, 15, 25, 35, 45. \( \sum f_i = 40 \) and \( \sum f_i x_i = 25 + 120 + 300 + 350 + 225 = 1020 \).
So \( \bar{x} = \dfrac{1020}{40} = 25.5 \).

Worked Example 2: Mode

For the same data the modal class is 20–30 (frequency 12). Here l = 20, h = 10, f1 = 12, f0 = 8, f2 = 10.
\( \text{Mode} = 20 + \dfrac{12 - 8}{24 - 8 - 10} \times 10 = 20 + \dfrac{40}{6} \approx 26.67 \).

Worked Example 3: Median

Cumulative frequencies of the same data are 5, 13, 25, 35, 40. Since n/2 = 20, the median class is 20–30 with cf = 13 and f = 12.
\( \text{Median} = 20 + \dfrac{20 - 13}{12} \times 10 = 20 + \dfrac{70}{12} \approx 25.83 \).

Points to Remember

1. Use class marks, not class limits, when computing the mean of grouped data.
2. The class width h must be the same for all classes in the mode and median formulas; if classes are unequal, check which class you are using.
3. The median class is decided by n/2, not by the highest frequency.
4. The mode and median formulas assume continuous classes; convert inclusive classes to true boundaries first.
5. Mean, median and mode of symmetric data coincide.