RBSE Class 10 Maths Notes Chapter 14 Statistics

These comprehensive RBSE Class 10 Maths Notes Chapter 14 Statistics will give a brief overview of all the concepts.

RBSE Class 10 Maths Chapter 14 Notes Statistics

Important Points

1. Central Tendency—Out of the given data, the data about which the maximum number of terms remain centred, is called the Central Tendency of data. It is also called measure of central tendency or mean. These are of three types :
(i) Mean
(ii) Mode
(iii) Median

2. Arithmetic Mean—If n values of any variable quantity are respectively x_1,x₂,x₃………..x_n, then their arithmetic mean will be

Here symbol Σ (sigma) has been used to represent the process of addition.

3. Mean—If the frequencies of the observations x_1, x₂, x₃ …….. x_n are respectively f_{1 ,}f₂, f₃ …… f_n then the sum of the values of all these observations =f₁x₁ +f₂x₂ +f₃x₃  …………. f_nx_n the number of observation is f₁ +f₂ +……….. +f_n. Then

Here the value of varies from 1 to n. This method is called Direct Method.
Arithmetic Mean from Ungrouped Frequency Distribution

Working Steps :

• Multiplying each variate with its frequency. Find the sum of (f_i x x.).
• In die sum we divide by the sum of frequencies.
• The quotient thus obtained will be the arithmetic mean.

4. Class-Mark or Mid-Point – The class-mark or mid-point of any class interval is found by finding the coverage of its upper limit and lower limit, i.e.,
\(\text { Class Mark }=\frac{\text { Upper Class Limit }+\text { Lower Class Limit }}{2}\)

5. Assumed Mean Method \((\bar{x})=a+\frac{\sum f_{i} d_{i}}{\sum f_{i}}\)
Where d_i = x. – a, a = Assumed Mean
∑f_i= N = Sum of Frequencies
Note—The assumed mean is taken that value of the variate (x) where frequency is maximum. By doing it mathematical computation becomes easier.

6. Step-Deviation Method — In this method we divide all the values of d_t = – a by a common number (say h). In this case we divide all the deviations by h and get new deviation in the form \(u_{i}=\frac{x_{i}-a}{h}\)

a = assumed mean
h = class size
∑f_i= N = sum of frequencies
Step deviation method will be convenient when all d_i have some common factor.

7. Mode – In the given observations the value of mode is that value which occurs most frequently i.e., the value of that observation where frequency is maximum is called the mode.
\(\text { Mode }=l+\left[\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right] \times h\)

Mode of grouped data is determined by this formula.
Here
l = lower limit of the modal class
h = size of class-interval
f₁= frequency of the modal class
f₀ = frequency of class preceding the modal class
f₂ = frequency of class succeeding the modal class.

8. Median

(A) If the’values of any variable quantity x are arranged in ascending or descending order, then the middle term of this series is called the median of the series.

(i) If the number of terms is odd in this case there will be only one middle term
\((M)=\left(\frac{n+1}{2}\right)^{\text {th }} \text { term }\)

(ii) If the number of terms is even, then.
\(\text { Median }(\mathbf{M})=\frac{n}{2} \text { th term }+\frac{\left(\frac{n}{2}+1\right)^{t h} \text { term }}{2}\)

(B) To find Median from ungrouped frequency distribution — The method of determining median from ungrouped frequency distribution is as follows :
(i) Prepare cumulative frequency table.
(ii) Find the value of \(\frac{\mathrm{N}}{2}\) where N = If.
(iii) The variate value with cumulative frequency just greater than \(\frac{\mathrm{N}}{2}\) will be the median.

(C) Median from grouped frequency distribution —
There are following points for finding median from grouped frequency distribution :
(i) Prepare cumulative frequency table.
(ii) Finding \(\frac{\mathrm{N}}{2}\) find the class-interval with cumulative frequency just greater than \(\frac{\mathrm{N}}{2}\).
(iii) Now for this class-interval calculate median with the help of the following formula:

\(\text { Median }=l+\left[\frac{\frac{n}{2}-c . f .}{f}\right] \times h\)
Here, l = lower limit of median class
n = number of observation
c.f = cumulative frequency of the class just preceding the median class
l= frequency of median class
h = class size

9. There exists the following relation between the three measures of central tendency
3 x Median = Mode + 2 x Mean

10. Cumulative frequency distribution can be represented graphically as a cumulative frequency curve or an ogive of the less than type or of the more than type.

11. The median of the grouped data is obtained by drawing a perpendicular from the points of intersection of both the types of ogives on the horizontal axis from the corresponding value of the point’of intersection with horizontal axis.