Statistics Class 10 Notes Maths: Chapter 14

Statistics

The three measures of central value are Mean, Median and Mode.
Mean is computed by the Direct method, Short-cut method and Step-deviation method.For each of the three methods, the class mark xi can be found by the following given formula : x1=1/2(lower limit + upper limit)
If a variable x takes the values x1, x2, x3,…, xn with the corresponding frequencies being f1, f2,…, fn respectively, then the arithmetic mean of these values is given by

$$(i) Mean =\frac{\Sigma (f_i × x_i)}{{\Sigma f_i}} [Direct Method]$$

$$\\(ii) Mean = A +\frac{\Sigma (f_i d_i)}{{\Sigma f_i}} [Assumed Mean Method/Short-cut Method], where$$
$$\\d_i=x_i – A [Axi = mid-value =\frac{lower limit + upper limit}{{2}}]$$

$$(iii) Mean = A +\begin{Bmatrix*}[r] h× \frac{\Sigma (f_i × U_i)}{{\Sigma f_i}}\end{Bmatrix*}[Step-Deviation Method],$$

$$\\ where h=class size and ui=\frac{X_i-A}{{h}}$$
$$\\ A=The Assumed Mean may be the class mark of the central class-interval.$$

The median of a distribution is the value of the variable which divides it into two equal parts. The median of a grouped or continuous frequency distribution may be computed by using the following formula :

where, l = lower limit of the median class*f = frequency of the median class h= width of the median class F = cumulative frequency of the class preceding the median class

$$\text{Median =}\text{l}+\frac{\frac{N}{2}-F}{f}×h$$

$$\text{and}\displaystyle\sum_{i=1}^n\text{f}_i$$

To find the Median class, one must first find out the value of N/2where N is the cumulative frequency of the given frequency distribution. The frequency of the class just greater thanN/2 givesthe Median class. The cumulative frequency (N) is given by the summation of all the frequencies in a grouped data.

Mode is the value of the variable with the maximum frequency. The mode of a continuous or grouped frequency distribution may be computed by using the following formula :

$$\text{Mode}=l+\frac{f – f_1}{2f – f_1 – f_2}×h$$

where, l = lower limit of the modal class** f = frequency of the modal class h= width of the modal class f1 = frequency of the class preceding the modal class f2 = frequency of the class following the modal class **In a frequency distribution, the Modal class is given by the class having the maximum frequency.