Statistics is a discipline of mathematics that deals with all aspects of data. Statistical knowledge aids in the selection of the most appropriate method for gathering data and the proper use of those samples in the analysis process to provide successful and effective results. Statistics is a branch of mathematics that deals with data gathering, organization, interpretation, analysis, and presentation. The primary goal of statistics is to organize the information gathered in terms of statistical surveys. Statistics is an important process since it assists in making data-driven decisions; hence, considerable knowledge of various statistical concepts is important. NCERT Solutions for Class 11 Maths Chapter 15 Statistics elaborates the types of statistics and various statistical concepts using illustrations and examples. There are 2 types of statistics discussed in NCERT Solutions for Class 11 Maths Chapter 15 Statistics:

• • • Descriptive Statistics: It is used to determine the variance, mean, and standard deviation of a sample set of collected data.

    • Inferential Statistics: It is a statistical method that uses sample data from a subset of a larger population. The results obtained from the sample data are then applied to the larger population data. The average weight of 50 students of a class, for example, is used to calculate the average weight of students throughout the standard.

 

The first and foremost concept of Class 11 Maths Chapter 15 Statistics is measures of dispersion. Dispersion is a statistical measure of data variability, scatter, or data spread. It measures how much the variables deviate from the central value. Dispersion is one of the most commonly utilized properties of data distributions. Dispersion measures are significant when comparing the consistency, reliability, and uniformity of variables or series. Class 11 Maths Chapter 15 Statistics highlights two types of measures of dispersion:

• • • Absolute measures of dispersion: When the various measures of dispersion are represented in the original unit of the variable or series. 

    • Relative measures of dispersion: When the various measures of dispersion are represented in ratios, percentage, and average. Relative measures of dispersion are also commonly known as coefficients of dispersion. 

 

The following measures of dispersion have been discussed in NCERT Solutions for Class 11 Maths Chapter 15 Statistics :

• • • Range 

    • Mean deviation 

    • Variance and standard deviation 

    • Frequency distribution

Range is an essential measure of dispersion, calculated as the difference between the maximum value and the minimum value. A lot of questions of Class 11 Maths Chapter 15 'Statistics' involve calculating the range first in order to perform other measures of dispersion. The mathematical formula to calculate the range of any series is as follows:

 

Range = Maximum value of the series - Minimum value of the series

 

The simplest measure of deviations from a value is mean deviation, and the value of mean deviation is usually calculated using a mean value. The mean deviation is calculated by subtracting the mean of the data from the actual values. So, the basic mathematical formula to calculate the mean deviation of the data is calculated as follows:

 

Mean deviation = Sum of all values of deviation from the mean / Total number of observations 

 

There are various merits to using mean deviation over other measures of dispersion:

• • • It is the least fluctuating measure of dispersion compared to other measures of dispersion like percentile range, quartile deviation, etc. 

    • It is the best measure of dispersion when you need to compare two or more series of data.

    • It is a preferred measure of dispersion since it takes into account all the values of the series.

Mean Deviation for Ungrouped Data 

The following are the steps to calculate the mean deviation for ungrouped data:

1. Calculate the mean of the ungrouped data 

2. Find the sum of deviations of actual absolute values from the mean, without minus sign and,

3. Divide the sum of absolute deviation values by the total number of observations.

 

Mean Deviation for Grouped Data

 

The frequency data can be differentiated in two ways:

• • • Grouping based on discrete values: The data contains discrete values and respective frequencies, and is grouped in a tabular framework.

    • Continuous data distribution: A specific range or class is given, and the data can take any value from the specified class. For example: In a continuous data class 25-30, data can take any value from 25 to 30. 

 

For both the categories, the steps for calculating the mean deviation are listed below:

 

Discrete distribution:

• • • Calculate the mean deviation by multiplying the distinct values with frequencies.

    • Find the mean. 

    • Subtract the multiplied values from the mean, and calculate the sum. 

    • Divide the result from the total of frequencies.

 

Continuous distribution:

• • • Assume that the frequency in each class is centred at its midway to get the mean deviation for the continuous frequency distribution. 

    • After you have found the midpoint, look for the mean deviation using the method similar to the discrete data distribution.

Limitations of Using Mean Deviation 

 

The limitations of using mean deviation are as follows:

• • • It is a little difficult to find the mean deviation of fractional values. 

    • The method of calculating mean deviation involves a lot of calculations, and so a lot of extra time.

    • As the negative signs are ignored in finding the mean deviation, it might not be the most reliable measure of dispersion.

 

The most important concepts of NCERT Solutions for Class 11 Maths Chapter 15 Statistics are variance and standard deviation as they are a true representation of data variability. Standard deviation is a measure of variability, whereas variance is a measure of how data points differ from the arithmetic mean. 

 

The variance is a measure of how far the values deviate from the mean and is denoted by σ2. Since the variance is square of deviations, it is always a non-negative value. 

 

The standard deviation helps us to calculate the degree of data deviation. Since it is the square root of the variance, it is also known as root mean square deviation.

Shortcut Method To Find Variance And Standard Deviation  

NCERT Solutions for Class 11 Maths Chapter 15 Statistics also provides a shortcut method to calculate the variance and standard deviation of the data by multiplying the data deviations with the respective frequencies.

The frequency of observation is defined as the number of times it occurs in a given set of data. The following are a few types of data and frequency distribution: 

• • • Discrete distribution: In this, the data is presented in such a way that the exact measurements of the units are available.

    • Continuous distribution: It refers to a frequency distribution that includes class groupings.

    • Cumulative distribution: It is the frequency derived by adding the frequencies of the previous classes. 

 

Frequency distribution is graphically analyzed using the following methods:

• • • Histogram

    • Pie charts 

    • Bar graphs

    • 'Less than' and 'More than’ graphs 

    • Frequency curves

 

The following are a few important questions and NCERT Solutions for Class 11 Maths Chapter 15 Statistics that will help you prepare better for the exams.

 

Q1. Find the variance and standard deviation of the following set of data: 57, 65, 73. 

 

Answer. To calculate the variance and standard deviation, we first need to find the mean of the given data. 

Mean = (57 + 65 + 73)/3 = 65  

Find the square of absolute deviation of the mean from actual values, which is equal to 8^2 + 0^2 + 8^2 = 128

 

So, Variance = 128/3 = 42.67

Standard deviation = square root of variance 

= 6.53

Q2. Find the range and mean of the following data: 13, 47, 26, 35, 93, 36, 12, 29, 82, 28.

 

Answer. Range = maximum value - minimum value

= 93 - 12 

= 81 

 

Mean = Sum of observations / Total number of observations 

= 401/10 

= 40.1 

You can try other questions related to Class 11 Maths Chapter 15 Statistics from NCERT Solutions for Class 11 Maths Chapter 15 Statistics by MSVGo. NCERT Solutions for Class 11 Maths Chapter 15 Statistics are provided by experienced subject experts keeping in mind the examination pattern.

The following are a few FAQs related to NCERT Solutions for Class 11 Maths Chapter 15 Statistics.

What are the different sub-topics of Chapter 15 Maths Class 11?

 

Ans. The different sub-topics of Class 11 Maths Chapter 15 are as follows:

• • • Measures of dispersion 

    • Mean deviation (both grouped and ungrouped)

    • Range

    • Standard deviation and variance, etc.

Are the NCERT Solutions for Class 11 Maths Chapter 15 helpful for students?

Ans. The NCERT Solutions for Class 11 Maths Chapter 15 Statistics offers a thorough explanation of the questions as well as a step-by-step strategy to tackle complex problems. Students will find it easier to grasp how to answer Class 11 Maths Chapter 15 Statistics problems after using NCERT Solutions for Class 11 Maths Chapter 15 Statistics by MSVGo.

 

What are the important formulae of Statistics Class 11?

 

Ans. Here are a few important formulae of Class 11 Maths Chapter 15 Statistics 'Statistics':

• • • Range = maximum value - minimum value

    • Mean = Sum of observations / Total number of observations

    • Variance = Square of absolute deviation of the mean from actual values / Total number of observations 

    • Standard deviation = Square root of the variance

    • Mean deviation = Sum of all values of deviation from the mean / Total number of observations