Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 15 Statistics will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 15 Statistics

**Measures Of Central Tendency:**

A measure of central tendency or average is a value, that is the representative of whole data and signifies its characteristics.

Different measures of central tendency are: (a) Mean (b) Median (c) Mode.

(a) Mean (Arithmetic Mean):

Mean of ungrouped data: The mean of ‘n‘ observations x_{1}, x_{2} …..x_{n} = \(\bar{x} \frac{\sum_{i=1}^n x_i}{N}\)

Mean of grouped data:

(i) Direct Method

If x_{i} are the mid values of the intervals with frequency f_{i} then the mean \(\bar{x}=\frac{1}{N} \sum_{i=1}^n f_i x_i\)

(ii) Shortcut Methods:

(1) Assumed mean method

Mean = \(\overline{\mathrm{x}}=\mathrm{A}+\frac{1}{\mathrm{~N}} \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{f}_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}\)

where A = the assumed mean ⇒ d_{i} = x_{i} – A

(iii) Step Deviation Method:

Mean = \(\overline{\mathrm{x}}=\mathrm{A}+\frac{\mathrm{C}}{\mathrm{N}} \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{f}_{\mathrm{i}} \mathrm{u}_{\mathrm{i}}\)

where A = The assumed mean, C = Class width

u_{i} = \(\frac{d_i}{C}=\frac{x_i-A}{C}\)

(b) Median

(i) Median of ungrouped data:

Let n is the number of observation.

Arrange the observations in ascending or descending order.

⇒ If n is odd, Median = \(\left(\frac{\mathrm{n}+1}{2}\right)^{\mathrm{th}}\) observation.

⇒ If n is even, Median = \(\frac{\left(\frac{\mathrm{n}+1}{2}\right)^{\mathrm{th}} \text { observation }+\left(\frac{\mathrm{n}}{2}+1\right)^{\mathrm{th}} \text { observation }}{2}\)

(ii) Median of grouped data

- Get \(\frac{\mathrm{N}}{2}\) and cummulative frequencies of all classes.
- Get the Median class.

Median class = The class whose cumulative frequency is just greater than (or near to) \(\frac{\mathrm{N}}{2}\).

Median = l + \(\frac{\mathrm{m}-\mathrm{c}}{\mathrm{fm}}\) × h,

where l = lower limit of median class .

h = Class width of median class M = \(\frac{\mathrm{N}}{2}\).

c = Cummulative frequency of the class preceeding the median class.

f_{m}= Frequency of the median class.

(c) Mode

Mode is the most frequent value.

⇒ We can find mode using the empirical formula:

Mode = 3 Median – 2 Mean.

(i) Mode for Grouped data

⇒ Get the Modal class: It is the class with maximum frequency.

Mode = \(l+\frac{\mathrm{f}_{\mathrm{m}}-\mathrm{f}_1}{2 \mathrm{f}_{\mathrm{m}}-\mathrm{f}_1-\mathrm{f}_2} \times \mathrm{c}\)

where l = lower limit of modal class.

f_{m} = Frequency of modal class.

f_{1} = Frequency of the class just preceeding modal class.

f_{2} = Frequency of the class just suceeding modal class.

**Measure Of Dispersion:**

The variability or scatter or spreading of data is known as dispersion.

Some of the measures of dispersion are:

(a) Range

(b) Mean deviation

(c) Variance

(d) Standard deviation

(a) Mean deviation: Mean deviation is the mean of absolute deviations of all observations from a central value (Mean or Median).

For Group – B

A = 35, C = 10

As C. V of Group – A is more, the data for group – A is more dispersed

For Group – A

**Analysis Of Frequency Distribution:**

Coefficient of variation (C. V) = \(\frac{\sigma}{x}\) × 100

Note:

- The distribution with greater C. V is more variable or dispersed and lesser C. V is less variable or more consistent.
- If two distributions have same mean then they can be compared on the basis of their standard deviation.