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Measures of Central Tendency

Generally, average value of a distribution in the middle part of the distribution, such type of values are known as measures of central tendency.

The following are the five measures of central tendency

1. Arithmetic Mean

2. Geometric Mean

3. Harmonic Mean

4. Median

5. Mode


Arithmetic Mean

The arithmetic mean is the amount secured by dividing the sum of values of the items in a series by the number.


1. Arithmetic Mean for Unclassified Data

If n numbers, x1, x2, x3,….., xn, then their arithmetic mean

Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce

2. Arithmetic Mean for Frequency Distribution

Let f1, f2 , fn be corresponding frequencies of x1, x2,…, xn. Then,

Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce

3. Arithmetic Mean for Classified Data

Class mark of the class interval a-b, then x = a + b / 2

For a classified data, we take the class marks x1, x2,…, xn of the classes as variables, then arithmetic mean

Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce

Step Deviation Method

Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce

where, A1 = assumed mean

ui = xi – A1 / h

fi = frequency

h = width of interval


4. Combined Mean

If x1, x2,…, xr be r groups of observations, then arithmetic mean of the combined group x is called the combined mean of the observation

A = n1 A1 + n2A2 +….+ nrAr / n1 + n2 +…+ nr

Ar = AM of collection xr

nr = total frequency of the collection xr


5. Weighted Arithmetic Mean

If w be the weight of the variable x, then the weighted AM

Aw = Σ wx / Σ w


Shortcut Method

Aw = Aw‘ + Σ wd / Σ w, Aw‘ = assumed mean

Σ wd = sum of products of the deviations and weight


Properties of Arithmetic Mean

(i) Mean is dependent of change of origin and change of scale.

(ii) Algebraic sum of the deviations of a set of values from their arithmetic mean is zero.

(iii) The sum of the squares of the deviations of a set of values is minimum when taken about mean.


Geometric Mean

If x1, x2,…, xn be n values of the variable, then

G = n√x1, x2,…, xn

or G = antilog [log x1 + log x2 + … + log xn / n]


For Frequency Distribution

Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce

or  Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce


Harmonic Mean (HM)

The harmonic mean of n items x1, x2,…, xn is defined as

Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce

If their corresponding frequencies f1, f2,…, fn respectively, then

Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce

The document Measures of Central Tendency | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Measures of Central Tendency - Mathematics (Maths) Class 11 - Commerce

1. What are measures of central tendency?
Ans. Measures of central tendency are statistical values that represent the center or average of a set of data. They provide a single value that summarizes the entire dataset, making it easier to understand and analyze the data. Common measures of central tendency include the mean, median, and mode.
2. How do you calculate the mean?
Ans. The mean is calculated by taking the sum of all the values in a dataset and dividing it by the total number of values. For example, if we have the numbers 2, 4, 6, and 8, the mean would be (2 + 4 + 6 + 8) / 4 = 5.
3. What is the median and how is it calculated?
Ans. The median is the middle value in a dataset when the values are arranged in ascending or descending order. If there is an odd number of values, the median is simply the middle value. If there is an even number of values, the median is the average of the two middle values. For example, in the dataset 2, 4, 6, 8, the median would be 5.
4. When is it appropriate to use the mode?
Ans. The mode is used when we want to find the most frequently occurring value or values in a dataset. It is particularly useful for categorical or nominal data, where the values represent categories or groups. For example, if we have a dataset of eye colors (blue, brown, green), the mode would tell us which eye color is the most common.
5. How do measures of central tendency help in data analysis?
Ans. Measures of central tendency provide a summary or average value of a dataset, making it easier to understand and compare different data points. They help in identifying the typical value or central point around which the data is distributed. This information can be useful in making predictions, identifying outliers, and drawing conclusions about the data.
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