Measures of Central Tendency Class 11 Revision Notes: This article hands out revision notes for CBSE Class 11 Chapter 5, Measures of Central Tendency. A PDF download link has also been attached to the article for your reference. Students can download the PDF link to save the notes for future use.
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Revision Notes for Class 11 Economics Chapter 5 are presented below:
What is Measures of Central Tendency?
Measures of Central Tendency is a numerical method to explain data in brief. It is commonly known as ‘averages’.
Statistical Measures of Central Tendency
There are three measures for the calculation of central tendency that are used most commonly:
- Arithmetic Mean
- Median
- Mode
1.Arithmetic Mean- Arithmetic mean is the most commonly used measure of central tendency. It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X= (X1+X2+X3.....Xn)/N
In general, if there are N observations as X1 , X2 , X3 , ..., XN , then the Arithmetic Mean is given by
Arithmetic mean is calculated by two methods:
Arithmetic mean for ungrouped data
It is calculated by the following methods:
- Direct Method- Arithmetic mean by direct method is the sum of all observations in a series divided by the total number of observations.
- Assumed Mean Method- Here you assume a particular figure in the data as the arithmetic mean on the basis of logic/experience. Then you may take deviations of the said assumed mean from each of the observations. You can, then, take the summation of these deviations and divide it by the number of observations in the data.
- Step Deviation Method- The calculations can be further simplified by dividing all the deviations taken from the assumed mean by the common factor ‘c’. The objective is to avoid large numerical figures, i.e., if d = X – A is very large, then find d'. This can be done as follows: d' = (X-A)/d. The formula is as follows: X= A+Σd'/N x C
Arithmetic mean for grouped data
For Discrete series
- Direct method- In the case of discrete series, frequency against each observation is multiplied by the value of the observation. The values, so obtained, are summed up and divided by the total number of frequencies. Symbolically,
- Assumed Mean method- Since the frequency (f) of each item is given here, we multiply each deviation (d) by the frequency to get fd. Then we get Σ fd. The next step is to get the total of all frequencies i.e. Σ f. Then find out Σ fd/Σ f. Finally, the arithmetic mean is calculated by X= A+ Σfd/Σf using the assumed mean method.
- Step Deviation Method- In this case, the deviations are divided by the common factor ‘c’ which simplifies the calculation. Here we estimate d' = d/c = (X-A)/c in order to reduce the size of numerical figures for easier calculation. Then get fd' and Σ fd'. The formula for arithmetic mean using the step deviation method is given as, X= A + (Σfd’/Σf )x c
For Continuous Series
Here, class intervals are given. The process of calculating the arithmetic mean in the case of continuous series is the same as that of a discrete series. The only difference is that the midpoints of various class intervals are taken. The direct method and step deviation method are followed for calculating continuous series, just like discrete series. The symbolical representation in both the cases remains same.
2.Median- The Median is the “middle” element when the data set is arranged in order of magnitude. Since the median is determined by the position of different values, it remains unaffected if, say, the size of the largest value increases. It can be easily computed by sorting the data from smallest to largest and finding out the middle value.
Discrete Series-In the case of discrete series the position of median i.e. (N+1)/2nd item can be located through cumulative frequency. The corresponding value at this position is the value of the median.
Continuous Series- In the case of continuous series you have to locate the median class where N/2th item [not (N+1)/2nd item] lies. The median can then be obtained as follows:
Median= L + (N/2-c.f)/f x h
Where, L = lower limit of the median class,
c.f. = cumulative frequency of the class preceding the median class, f = frequency of the median class,
h = magnitude of the median class interval.
Quartiles- Quartiles are the measures that divide the data into four equal parts, each portion contains an equal number of observations. There are three quartiles. The first Quartile (denoted by Q1 ) or lower quartile has 25% of the items of the distribution below it and 75% of the items are greater than it. The second Quartile (denoted by Q2 ) or median has 50% of items below it and 50% of the observations above it. The third Quartile (denoted by Q3 ) or upper Quartile has 75% of the items of the distribution below it and 25% of the items above it.
Percentiles- Percentiles divide the distribution into hundred equal parts, so you can get 99 dividing positions denoted by P1, P2, P3, ..., P99. P50 is the median value.
Calculation of Quartiles
The method for locating the Quartile is the same as that of the median in the case of individual and discrete series. The value of Q1 and Q3 of an ordered series can be obtained by the following formula where N is the number of observations:
Q1 = size of (N+1)/4 item
Q2 = size of 3(N+1)/4 item
3.Mode- Mode is the most appropriate measure. It is repeated the highest number of times in the series. Mode is the most frequently observed data value. It is denoted by
Computation of Mode
Discrete Series- Simply, pick the number that has more frequency.
Continuous Series- In the case of continuous frequency distribution, the modal class is the class with the largest frequency. The mode can be calculated by using the formula:
Mo = L+ (D1 / D1+D2) x H
Where L = lower limit of the modal class D1 = difference between the frequency of the modal class and the frequency of the class preceding the modal class (ignoring signs). D2 = difference between the frequency of the modal class and the frequency of the class succeeding the modal class (ignoring signs). h = class interval of the distribution.
Note: The median is always between the arithmetic mean and the mode.
To download the revision notes for Class 11 Economics Chapter 5, click on the link below
| Download Revision Notes for CBSE Class 11 Economics Chapter 5 |
This chapter is comparatively short and easier to grasp. These notes shall be enough for your preparation for annual examinations.
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