The compilation of these Measures of Central Tendency Notes makes students exam preparation simpler and organised.
Calculation of Mean, Median, and Mode
When working on a given set of data, it is not possible to remember all the values in that set. But we require inference of the data given to us. This problem is solved by mean median and mode. These, known as measures of central tendency, represent all the values of the data. As a result, they help us to draw an inference and an estimate of all the values. Let us learn about the measures of central tendency and the calculation of Mean Median and Mode.
The Measures of Central Tendency – Mean Median and Mode
As already discussed, the mean median and mode are known as measures of central tendency. They are also known as statistical averages. Their simple function is to mathematically represent all the values in a particular set of data. Hence, this representation shows the general trend and inclination of all the values.
An average provides a simple way of representing all the individual data. It also aids in the comparison of different groups of data. In addition to this, an average in economic terms can represent the direction an economy is headed towards. Hence, it can be easily used to formulate policies and bring about reform for a better economy.
For example, a low per capita income is an indication for the government to formulate policies focused on the increase in income of people. We will learn about three measures of central tendency- Mean Median and Mode.
Arithmetic Mean
The first concept to understand from Mean Median and Mode is Mean. Mean is simply defined as the ratio of the summation of all values to the number of items. Note that there are two types of arithmetic mean which are simple arithmetic mean and weighted arithmetic mean.
Mean = ∑X ÷ N
Here, ∑X = Sum of all the individual values and
N = Total number of items
The simple arithmetic mean considers all the values in data as equal and grants equal importance to each value. Whereas, in weighted arithmetic mean weights or importance is assigned to the values. Without further delay let us study methods of constructing mean.
Mean Calculation for Individual Series
1. Direct Method
In this method, the formal definition of mean is used. The values of items are simply summed and divided by the number of observations.
Mean = ∑X ÷ N
2. Assumed Mean Method
In the assumed mean method, a value is randomly selected as an assumed mean. Generally, the value is around the centre of the series as this facilitates calculations( the calculated deviations are both negative and positive around the assumed value, hence they cancel out or sum up to a very small value).
The assumed mean is found by dividing the maximum and minimum values by 2. Now deviation of each value from the assumed mean is calculated as deviation = value of the item – assumed value of mean in a separate column. The summation of these deviations are calculated and the actual mean is derived using the given formula:
Mean = A + (∑d ÷ N)
Here, A = Assumed value of the mean
∑d = Summation of deviations and
N = Number of observations
Mean Calculation for Discrete Series
Again in the case of discrete series, the mean can be calculated by three approaches as follows:
1. Direct Method
The formula for the direct method is as follows:
Mean = ∑fX/∑f
Here, ∑fX = Summation of the product of values of items with their corresponding frequencies
∑f = Summation of all the frequencies
2. Assumed Mean Method
The basic idea behind the assumed mean method remains the same for discrete series too, but the overall formula changes a bit to incorporate the addition of frequencies assigned to observations as follows:
Mean = A + ∑fd/∑f
Here, A = Assumed mean value
∑fd = Summation of the product of deviations and corresponding frequencies
∑f = Summation of the frequencies
3. Step Deviation Method
Similar to the assumed mean method, the concept behind step deviation method is to make calculations easier. It is a simpler variant of the assumed mean method and is used when there is a common factor among all the deviations by which they can be divided to reduce their values.
The factor is indicated by ‘C’. The deviation, when reduced by this factor, is known as a step-deviation. The formula is as follows:
Mean = A + (∑fd’/∑f) × C
C = The common factor using which deviations are converted to step-deviations
Note: In this method step-deviation denoted by d’ is used and not d.
d’ = (X – A)/C
Here, X = The value of the item
A = Assumed value of mean and
C = Common factor chosen
Mean Calculation for Frequency Distribution
The three approaches towards calculating mean for frequency distribution series are as follows:
1. Direct Method
In a frequency distribution, instead of individual values of observations, classes are mentioned. Hence to find the mean we need a single value that can represent the interval.
Such a value is found by adding the upper and lower class values and dividing the sum by 2. This value is known as mid-value. It is usually represented by m or Xi. Therefore the formula for calculating mean by direct method for frequency distribution is:
Mean = ∑fXi/∑f
OR
Mean = ∑fm/∑f
Here, ∑fXi or ∑fm = Summation of the product of mid values and corresponding frequencies
∑f = Summation of the frequencies
2. Assumed Mean Method
The overall idea of the assumed mean method here also remains the same except the fact that the concept of mid values is incorporated. The formula remains the same as the following:
Mean = A + ∑fd/∑f
Here, A = assumed mean value
∑fd = Summation of the product of deviation and corresponding frequency
∑f = Summation of the frequencies
Note: Here mid-values of the classes are used for all calculation purposes.
3. Step-Deviation Method
Since the assumed mean method, in this case, is almost the same, likewise, the step deviation method also remains almost the same as it is a simpler version of the former. The only change incorporates the concept of mid-values. The formula is as follows:
Mean = A + (∑fd’/∑f) × C
Weighted Arithmetic Mean
As already mentioned weighted arithmetic mean assigns weights to observations depending on their importance. Different items of the series are weighted according to their relative importance and the mean is hence called weighted arithmetic mean. The formula for calculation of weighted mean is mentioned below:
Mean = ∑WX/∑W
Here, ∑WX = Summation of the product of items and the corresponding weights assigned to them
∑W = Summation of the weights
Median
Another measure of central tendency i.e. (Mean Median and Mode) is median which is essentially known as the central value of a series. Median is a value in series such that it divides the series exactly in halves. This means one half of the series above-median contains all values greater than it and the other half contains all values smaller than the median. Hence median is the mid-value.
Calculation of Median
Median for Individual series
In individual series, where data is given in the raw form, the first step towards median calculation is to arrange the data in ascending or descending order. Now calculate the number of observations denoted by N. The next step is decided by whether the value of N is even or odd.
- If the value of N is odd then simply the value of (N+1)/2 th item is median for the data.
- If the value of N is even, then use this formula:
Median = [size of (N+1)/2 term + size of (N/2 + 1)th term] ÷ 2
Median for Discrete Series
The first step for calculation of median here also involves arranging the data in ascending or descending order. This is followed by the conversion of simple frequencies into cumulative frequencies. Hence another column for cumulative frequency needs to be constructed, wherein the last value is labeled as the value of N (i.e ∑f).
Next, we need to find the value of (N+1)/2. Lastly, the value corresponding to the cumulative frequency just greater than (N+1)/2 is termed as the median for the data.
Median for Frequency Distribution
As in all other types of distributions, here also initially we arrange the classes in either ascending or descending order. Next, we need to find the cumulative frequencies. The last value in the cumulative frequency column which is ∑f is labeled as N. This is followed by the calculation of the value of N/2.
Further, the class corresponding to the cumulative frequency just greater than this value is known as the median class. Lastly, the median value is calculated by applying the following formula:
Median = l/2 + h/f[N/2 – C]
Here, l = The lower limit of the median class
h = size of the class
f = Frequency corresponding to the median class
N = Summation of frequencies
C = The cumulative frequency corresponding to the class just before the median class
Mode
Now we come to the third concept of Mean Median and Mode. It is the measure of central tendency aims at pointing out the value that occurs most frequently in a series. This value, when it represents the data is known as the mode of the series. Mode simply refers to the value that occurs the maximum number of times in a distribution.
Calculation of Mode
Mode for Individual Series
In the case of individual series, we just have to inspect the item that occurs most frequently in the distribution. Further, this item is the mode of the series.
Mode for Discrete Series
In discrete series, we have values of items with their corresponding frequencies. In essence, here the value of the item with the highest frequency will be the mode for the distribution.
Mode for Frequency Distribution
Lastly, for frequency distribution, the method for mode calculation is somewhat different. Here we have to find a modal class. The modal class is the one with the highest frequency value. The class just before the modal class is called the pre-modal class. Whereas, the class just after the modal class is known as the post-modal class. Lastly, the following formula is applied for the calculation of mode:
Mode = l + h [(f1 – f0)/(2f1 – f0 – f2)]
Here, l = The lower limit of the modal class
f1 = Frequency corresponding to the modal class
f2 = Frequency corresponding to the post-modal class
and f0 = Frequency corresponding to the pre-modal class
Example:
Question:
Calculate mode for the following data:
Answer:
As the frequency for class 30 – 40 is maximum, this class is the modal class.
Classes 20 – 30 and 40 – 50 are pre-modal and post-modal classes respectively.
The mode is:
Mode = 30 + 10 × [(15 – 10)/(2 × 15 – 10 – 10)]
= 30 + 5
= 35
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