Statistics Class 10 ||Maths|| Chapter 13 NCERT Notes

1. Introduction to Statistics:

Statistics is the branch of mathematics that deals with data collection, organization, analysis, interpretation, and presentation. In Class 10, the focus is on measures of central tendency and the study of grouped data.

2. Measures of Central Tendency:

The measures of central tendency are used to represent a dataset with a single value that describes the center of the data. The main measures covered in this chapter are:

Mean
Median
Mode

These are applied to grouped data in this chapter.

3. Mean of Grouped Data:

The mean is the arithmetic average of all data points. For grouped data, the mean can be calculated using three methods:

i) Direct Method:

This method is the simplest, where the mean is computed using the formula:

Mean (\overset{ˉ}{x}) = \frac{\sum f_{i} x_{i}}{\sum f_{i}}

Where:

$f_{i}$ = frequency of the ith class
$x_{i}$ = mid-point of the ith class (class mark)
$\sum f_{i}$ = total frequency

Steps:

Calculate the mid-point $x_{i}$ of each class interval.
Multiply $x_{i}$ by $f_{i}$ .
Find the sum of all $f_{i} x_{i}$ .
Divide by the total frequency $\sum f_{i}$ .

ii) Assumed Mean Method:

In cases where the direct method involves large numbers, the assumed mean method simplifies calculations. The formula is:

Mean (\overset{ˉ}{x}) = A + \frac{\sum f_{i} d_{i}}{\sum f_{i}}

Where:

$A$ = assumed mean (an approximation chosen from the mid-points)
$d_{i} = x_{i} - A$ , the deviation of $x_{i}$ from the assumed mean

Steps:

Assume a mean $A$ from the mid-points of the class intervals.
Calculate deviations $d_{i} = x_{i} - A$ .
Multiply $d_{i}$ by $f_{i}$ to get $f_{i} d_{i}$ .
Sum up $f_{i} d_{i}$ .
Divide by the total frequency and add $A$ .

iii) Step Deviation Method:

This method is used when the class intervals are large or vary significantly. It further simplifies calculations using step deviations. The formula is:

Mean (\overset{ˉ}{x}) = A + (\frac{\sum f_{i} u_{i}}{\sum f_{i}}) \times h

Where:

$u_{i} = \frac{x_{i} - A}{h}$ (standardized deviation)
$h$ = class size (difference between consecutive class boundaries)
$A$ = assumed mean

Steps:

Assume a mean $A$ from the mid-points.
Calculate $u_{i} = \frac{x_{i} - A}{h}$ .
Multiply $u_{i}$ by $f_{i}$ .
Sum up $f_{i} u_{i}$ .
Calculate the mean using the formula above.

4. Mode of Grouped Data:

The mode is the value that occurs most frequently in a data set. For grouped data, it is calculated using the formula:

Mode = l + (\frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}}) \times h

Where:

$l$ = lower boundary of the modal class
$f_{1}$ = 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
$h$ = class size

Steps:

Identify the modal class (the class with the highest frequency).
Use the formula to find the mode.

5. Median of Grouped Data:

The median is the value that divides the dataset into two equal halves. For grouped data, the median is calculated using the formula:

Median = l + (\frac{\frac{N}{2} - C_{f}}{f}) \times h

Where:

$l$ = lower boundary of the median class
$N$ = total frequency
$C_{f}$ = cumulative frequency of the class preceding the median class
$f$ = frequency of the median class
$h$ = class size

Steps:

Calculate $\frac{N}{2}$ , where $N$ is the total frequency.
Identify the median class (the class where cumulative frequency is greater than $\frac{N}{2}$ ).
Use the formula to find the median.

6. Cumulative Frequency Distribution:

Cumulative frequency refers to the running total of frequencies. There are two types of cumulative frequency distributions:

Less than type: Shows the total number of observations less than or equal to a particular value.
More than type: Shows the total number of observations greater than or equal to a particular value.

Cumulative Frequency Curves (Ogive):

Ogive is a graphical representation of the cumulative frequency distribution.
Two types of ogives can be drawn:
- Less than Ogive: Plot cumulative frequency against the upper boundary of the class intervals.
- More than Ogive: Plot cumulative frequency against the lower boundary of the class intervals.

The median can be determined graphically using the intersection of the two ogives.

7. Summary of Key Formulas:

Mean (Direct Method):
$Mean = \frac{\sum f_{i} x_{i}}{\sum f_{i}}$
Mean (Assumed Mean Method):
$Mean = A + \frac{\sum f_{i} d_{i}}{\sum f_{i}}$
Mean (Step Deviation Method):
$Mean = A + (\frac{\sum f_{i} u_{i}}{\sum f_{i}}) \times h$
Mode:
$Mode = l + (\frac{f_{1} - f_{0}}{2 f_{1} - f_{0} - f_{2}}) \times h$
Median:
$Median = l + (\frac{\frac{N}{2} - C_{f}}{f}) \times h$

8. Graphical Representation of Cumulative Frequency:

Ogives are used to represent cumulative frequency data.
The intersection of the “less than” and “more than” ogives gives the median.

9. Example Problems:

Let’s consider an example to solidify the concepts. Suppose you are given the following frequency distribution:

Class Interval	Frequency
10-20	5
20-30	8
30-40	12
40-50	15
50-60	10

For this data:

Compute the mean using all three methods.
Find the mode of the data.
Calculate the median.

This was a detailed overview of Chapter 13 - Statistics. If you need help with any specific examples or explanations of the graphical method, feel free to ask! ✒️✨

studyboost.co

Search This Blog