CBSE · Class 10 · Mathematics · Chapter 13

Statistics — Formula Sheet

Board Formulas

11 formulas

1.Class Mark
$x_{i} = \frac{upper class limit + lower class limit}{2}$
The mid-point of a class. When finding the mean, each class is represented by its class mark.
2.Mean: Direct Method
$x ˉ = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$
$x ˉ$ : Mean of the data · $f_{i}$ : Frequency of the i-th class · $x_{i}$ : Class mark of the i-th class
Multiply each class mark by its frequency, add, and divide by the total frequency. Best when fᵢ and xᵢ are small numbers.
3.Mean: Assumed Mean Method
$x ˉ = a + \frac{\sum f _{i} d _{i}}{\sum f _{i}}, d_{i} = x_{i} - a$
$a$ : Assumed mean (one of the class marks) · $d_{i}$ : Deviation of the class mark from a
a is the assumed mean, usually a class mark near the centre of the table. The dᵢ are smaller than the xᵢ, so the products are easier.
4.Mean: Step-Deviation Method★
$x ˉ = a + h (\frac{\sum f _{i} u _{i}}{\sum f _{i}}), u_{i} = \frac{x _{i} - a}{h}$
$h$ : Class size (common factor of the dᵢ) · $u_{i}$ : Step deviation of the i-th class
Use when all the dᵢ have a common factor h, usually the class size. All three methods give the same mean.
5.Modal Class
$modal class = class with the maximum frequency$
The mode lies in this class. Find it before applying the mode formula.
6.Mode of Grouped Data★
$Mode = l + (\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}) \times h$
$l$ : Lower limit of the modal class · $h$ : Class size · $f_{1}$ : 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
f₀ and f₂ are the frequencies of the classes just before and just after the modal class. If the modal class is the first class, f₀ = 0.
7.Cumulative Frequency (Less Than Type)
$c f_{k} = \sum_{i = 1}^{k} f_{i}$
Running total of the frequencies up to and including the k-th class. The last cumulative frequency equals n = Σfᵢ.
8.Median Class
$median class = class whose c f is greater than and nearest to \frac{n}{2}$
n = Σfᵢ is the total frequency. Compare n/2 with the cumulative frequency column, not the frequency column.
9.Median of Grouped Data★
$Median = l + (\frac{\frac{n}{2} - c f}{f}) \times h$
$l$ : Lower limit of the median class · $n$ : Total frequency, Σfᵢ · $c f$ : Cumulative frequency of the class preceding the median class · $f$ : Frequency of the median class · $h$ : Class size
cf is the cumulative frequency of the class PRECEDING the median class, and f is the frequency of the median class itself.
10.Empirical Relationship★
$3 Median = Mode + 2 Mean$
An approximate relation between the three measures of central tendency. Use it when two measures are known and the third is asked.
11.Making Classes Continuous
$adjustment = \frac{1}{2} [(lower limit of next class) - (upper limit of a class)]$
For classes 10–19, 20–29, … the gap is 1, so subtract 0.5 from every lower limit and add 0.5 to every upper limit: 9.5–19.5, 19.5–29.5, … The class size stays 10.