Statistics Class 11 Study Notes

Mathematics is a subject that cannot be mastered by just reading the concepts and definitions; it needs intense and consistent practice to accomplish excellent results. So, in this article, we will be going through an important topic known as Statistics. The word “Statistics” refers to the collection, classification, analysis, and representation of data. It helps in gaining valuable insights from the data and thus in making beneficial decisions based on it. Over the academic years, you must have come across terms like mean, median, mode, standard deviation, correlation, etc. Statistics class 11 is an advanced and comprehensive version of these topics. Read this blog to understand statistics class 11 completely!

The basics – Mean, Median, and Mode.

First, let us understand the fundamental terminologies used in statistics class 11:

The mean of a particular data set is the summation of all the quantities divided by the number of quantities. Whereas, the median can be thought of as the middle value of the data set. To calculate the median, the data set should be arranged in increasing order. Lastly, a mode is a value that occurs the most number of times. 

Example: Suppose we have a collection of numbers such as {2, 2, 3, 4, 5, 8}, let us find the mean, median, and mode for this.

Mean = sum of all the quantities/ number of quantities

Therefore, Mean = {2+2+3+4+5+8}/6 = 4

Likewise, median = (n+1)/2 th number =6/2th number = 3 (for odd number of values, for even it is (Mean of n/2 th and (n/2) + 1 th number) and the mean = 2 as it is occurring the most number of times.  Let us understand the above-explained concept with the help of an example.

Example: Suppose we have been provided with the marks of students of a class as given below, 36, 72, 46, 42, 60, 45, 53, 46, 51, 49, 42, 30, 42, 38. We now have to calculate the mean, median, and mode of the data set given above.

Let us first arrange the data in ascending order, by doing so we get 30, 36, 38, 42, 42, 42, 45, 46, 46, 49, 51, 53, 60, and 72.

 Here number of observations or n = 14 which is even.

As discussed mean = = sum of all the quantities/ number of quantities = (30 + 36 + 38 + 42 + 42 + 42 + 45 + 46 + 46 + 49 + 51 + 53 + 60 + 72) / 14 = 652 / 14

Therefore, Mean = 46.57

Now, median is the average of 7th and 8th observations that is 

Median = (45 + 46) / 2 = 45.5

Lastly, Mode = 42 (as it is occurring the most number of times)

After studying Statistics class 11, you will have a clear understanding of how different it is from core mathematics. If you want to know more about it, read our blog on the difference between statistics and mathematics and discover careers that are specific to each of these fields! 

Measures of Dispersion 

For estimating the scatter or dispersion in data, there are four ways mentioned in Statistics class 11 by which we can gather that:

Range

The difference between the maximum and minimum values of each group gives us the range. So, for the previous example, we can calculate range as, Range = 8-2 =6. 

Mean Deviation 

Suppose we have a number say, y, so the deviation of that observation y from a fixed value say, b will be y-b. To observe the scattering of values of y from a central value b, we have to obtain the deviations about b. An absolute measure of dispersion is the mean of these deviations. 

Now, to get the mean, we need to get the sum of the deviations. But, a measure of central tendency lies between the maximum and the minimum values of the given data set. Hence, some of the deviations will be negative and some positive. So, the sum of deviations may disappear. Moreover, the sum of the deviations from mean (y) is zero.

Thus, for determining the measure of the dispersion from a fixed number say, x, we need to get the mean of the absolute values of the deviations from the central value. This means is known as the ‘mean deviation’. 

M.D. (mean deviation) = Sum of absolute values of deviations from x/ Number of observations

Let us see this with an example mentioned in the chapter Statistics class 11. Suppose we have the following data and we have to calculate the mean deviation for this data.

Firstly we will calculate the mean of this data as shown below.

Now, using the mean we will find the absolute values of deviations from the mean which are 6, 3, 2, 1, 0, 2, 3, 7. Now, the required mean deviation is

Mean deviation for ungrouped data

For n observation {x1, x2, …, xn} the mean deviation about their mean is given by-

M.D. = sum of absolute values of the deviations (xi- the calculated mean)/
number of observations (n)

While the mean deviation about their median = M will be,

M.D. = sum of absolute values of the deviations (xi- M)/ number of observations (n).

We will see this through an example below. 

First, we will arrange the dataset in ascending order, for finding out the median of the given dataset.

We now can use this value of median to calculate mean deviation about the median.

Mean deviation for grouped data

For n discrete observations {x1, x2, … , xn} occurring with frequencies {f1, f2,…fn} respectively. The mean deviation about their mean will be

The mean deviation about their Median = M is

Where xi is the midpoints of the classes, x¯ and M are the mean and median of the distribution, respectively.

Variance 

Another topic mentioned in Statistics class 11 is Variance. It is the measure of data spread and mathematically is calculated as the average of the squared differences from the mean. The symbol of variance is sigma squared (σ2). To calculate the variance, first, we take out the mean and then for each observation, subtract the mean and square the result.

              σ2= (x1-mean) ^ 2 +(x2-mean) ^ 2 + . . . + (xi-mean) ^ 2 / number of observation

Let us try visualizing this via an example. Suppose we have to find the variance of a frequency distribution as shown in the image below.

Before we start calculating variance, we have to first find out the mean of this distribution. This can be done by finding the midpoint i.e. xi and the product of the midpoint and frequency as shown below.

We now  have all the required parameters for calculating the mean, which can be calculated as shown below.

Now let us find the difference between the mean and each observation and calculate the sum of all the results.

Now we have all the variables for finding out the variance, so let us calculate it.

Also Read: Class 11 Maths Syllabus

Standard Deviation 

In easy terms, the standard deviation is the square root of variance. If the standard deviation is low, the values tend to be close to the mean of the set, while if the standard deviation is high; the values are away from the mean. The symbol of variance is sigma (σ) and can be calculated by square rooting the value of variance. 

The standard deviation for discrete frequency distribution is when the data is given with their frequencies and is calculated by:

where, N = number of observations, fi = different values of frequency f and xi =different values of x

Seeing this by using an example will help us understand it better. We have been provided with the following data as shown in the image below and we have to find the standard deviation.

We will now calculate all the necessary parameters for finding out the standard deviation mentioned in Statistics Class 11.

Now we have all the values necessary for determining the standard deviation, and can be obtained as shown in the following image.

Therefore, standard deviation = 6.12

While the standard deviation for a continuous frequency distribution or grouped data is calculated by- 

where,N = number of observations, fi = different values of frequency f, xi = different values of mid points for ranges and x¯= mean for the ranges

For finding the standard deviation for a continuous frequency distribution you can calculate the variance and square root the result ( refer to the variance example and you can see the formula for variance is the same as standard deviation except for the square root).

Coefficient of Variation (CV)

Coefficient of variation, also called relative standard deviation, is defined as the ratio of the standard deviation to the mean. It is denoted as a percentage, and it tells us about the degree of scattering around the mean. The higher the value of CV, the more is the scattering, while low CV denotes less dispersion.

Coefficient of Variation = (Standard deviation / Mean) * 100

To understand this Statistics Class 11 concept, let us go through an example below.

For comparing the variation, we have to calculate the coefficient of variation using the formula explained above for each subject as shown below.

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After calculating the coefficient of variation, we can compare which subject shows the highest variability in marks as well as which subject shows the lowest.

Statistics Class 11 Practice Questions

You can also practice some questions (see images below) for mastering all the concepts as solving these questions will improve both your accuracy and speed.

Statistics class 11 is one of the very vast and fundamental topics of Mathematics but also a fun topic when learnt by heart. We hope that through this article, we have cleared your concepts and assisted you in understanding better how Statistics work. If you want similar explanations and insights into various other subjects, please do visit the Leverage Edu. You will find simple explanations for subjects such as Biology, Chemistry or even Short Math tricks to get around difficult problems and learn new topics.