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Statistics

Range and Mean Deviation for Grouped Data

Most of the data we deal with in real life is in a grouped form. The amount of data is generally large and is associated with corresponding frequencies (sometimes we divide data items into class intervals). For example- we are given data about people of varying age groups in a library. The data will be displayed as classified into intervals associated with their frequencies depending on the number of people belonging to each interval. Here, we will be studying methods to calculate range and mean deviation for grouped data.

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Grouped Data: Further Classification

Grouped data can be further classified into two types. These are:

  • Discrete frequency distribution- In this type, the individual data members are accompanied by their corresponding frequencies. Effectively there are two columns. One column is for the individual data items denoted by and other column consists of frequencies denoted by f.
  • Continous frequency distribution- In this classification, the data members are grouped into various class intervals and are associated to their corresponding frequencies. In essence, there is one column for class intervals and another column for frequencies.

Range and Mean Deviation of Grouped Data

Discrete Frequency Distribution

Range

As we know already that range is defined as the difference between the maximum and minimum value. Here also we select the largest value denoted by and smallest value denoted by S. Thereafter range is calculated as Range= L-S.

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Mean Deviation

Mean deviation calculates the dispersion of data items around a measure of central tendency( generally taken as median or mode). So before calculating mean deviation, we need to calculate mean or median according to the need of the problem.

Learn more about Range and Mean Deviation in detail here.

Mean for Discrete Frequency Distribution

Mean is calculated as the ratio of the summation of the product of data items with their corresponding frequency to the summation of frequency. Mathematically:

Mean= ∑fX÷N

Here, ∑fX= Summation of the product of data members (X) with their corresponding frequencies (f) and N= Summation of the frequencies(∑f)

Median for Discrete Frequency Distribution

Firstly, grouped data is arranged in ascending or descending order (mostly ascending order). Then, A separate column for cumulative frequency is constructed. Further, we note whether the value of summation of frequency or the last value of cumulative frequency column is even or odd. This value is denoted as N.

If N is odd then we calculate N/2. If N is even then we calculate: [(N/2) + (N/2 +1)]/2 . This calculated value, in either case, is compared to the cumulative frequencies. The observation whose corresponding cumulative frequency is just greater than this value is termed as the median.

Learn about Variance and Standard Deviation here.

Mean Deviation Calculation

The calculated mean or median is denoted by ‘a’. We calculate the absolute value of the deviation of each observation from ‘a’. Further, we find the summation of the product of deviations for each observation with their corresponding frequency. Lastly, this is divided by the summation of frequency column termed as N.

Mean Deviation= (∑ fD)/N

Here, ∑fD= Summation of the product of the absolute value of deviation (D, calculated as |X-a|) with the corresponding frequency (f)

Continuous Frequency Distribution

Range

In case of continuous frequency distribution, range, according to the definition, is calculated as the difference between the lower limit of the minimum interval and upper limit of the maximum interval of the grouped data. That is for X:  0-10, 10-20, 20-30 and 40-50, range is calculated as 40-0=40.

Mean Deviation

Let’s recall the mathematical formulae for calculating mean and median for the continuous frequency distribution of grouped data

Median for continuous Frequency Distribution

The calculation of median commences with arranging the grouped data generally in ascending order. After this, we make a separate column consisting of cumulative frequencies. The summation of frequencies or the last value is denoted as N. Then we proceed to calculate (N/2).

Now we check the cumulative frequency which is just greater than this value and the class it corresponds to. This class is noted as the median class. After deducing the median class, we apply the following formula:

Median= l + [ (N/2 – C)/f]/h

Here, C= Cumulative frequency corresponding to the class just before the median class and h= Size of the class intervals

Mean for Continuous Frequency Distribution

According to the definition, calculation of mean in this case also remains the same as in the case of the discrete frequency distribution of grouped data. The only difference is that the mid-points of each class denoted by Xi [ Xi= (h+l)/2]  are used to represent the class. Mean is calculated:

Mean= ∑fXi/N

Here, ∑fXi= Summation of the product os mid-points of each class with the corresponding frequency

Calculation of Mean Deviation

The calculation of mean deviation here proceeds in the same manner as for discrete frequency distribution except for the introduction of the concept of mid-points. For ease of calculation, it is assumed that the frequency of a class is centered around the mid-point of the interval. The mid-point is calculated as : (sum of upper and lower limit of the class)/2. The following formula is used:

Mean Deviation= (∑ fD)/N

Shortcut method or Step-Deviation method

Note: This method helps in finding mean deviation about mean only

The step-deviation method is used to easily calculate the mean compared to the traditional method. This method makes use of the concept of assumed mean to find out the actual mean. What is assumed mean? For the purpose of ease of calculations, a particular observation(or mid-point) is assumed to be the mean and deviations are recorded about this assumed mean. Later the actual mean is found out.

Note that the step deviation aids in finding only the mean. The remaining approach for calculation of mean deviation remains same as explained above. The assumed mean and factor by which the deviations are reduced are denoted by ‘m’ and ‘c’ respectively. Deviations are calculated as : D’= (Xi-m)/c. Lastly, the actual mean is calculated as :

Actual Mean= m + (∑fD’/N)×c

Here, m= The assumed mean

D’= The step deviations

c= The common factor by which deviations are reduced

Learn Range and Mean Deviation for Ungrouped Data here.

A Solved Example for You

Question 1: Find the mean deviation about mean for following data. Use step-deviation method.

Marks obtained 10-20 20-30 30-40 40-50 50-60 60-70 70-80
Number of students 2 3 8 14 8 3 2

Answer : Here the assumed mean

Marks obtained Number of students(f) Mid-points(Xi) D’=(Xi-45)/10 fD’  |Xi – a|  f|Xi – a|
10-20 2 15 -3 -6 30 60
20-30 3 25 -2 -6 20 60
30-40 8 35 -1 -8 10 80
40-50 14 45 0 0 0 0
50-60 8 55 1 8 10 80
60-70 3 65 2 6 20 60
70-80 2 75 3 6 30 60
∑=40 ∑=0 ∑=400

The assumed mean(m) and common factor(c) are taken to be 45 and 10 respectively.

Actual mean (a)= 45 + (0/40)×10 = 45

So, mean deviation = 400/40= 10

Question 2: What is grouped and ungrouped data?

Answer: First of all, both these forms are very useful forms of data. Group data refers to a form of data that has been organized into groups from the raw data. On the other hand, ungrouped data is a form of data that is raw, which means that it has only been collected and not sorted.

Question 3: What is grouped continuous data?

Answer: It refers to data that can take any values For instance; it can include time, weight, and height. Moreover, because if its capability to take any value, there are an infinite number of possible outcomes. Furthermore, we need to group continuous data before they can be represented in a frequency table or statistical diagram.

Question 4: How to find the range?

Answer: Range refers to a set of data that is the difference between the highest and the lowest values in the set. For finding the range, of a set, first of all, order data from least to greatest. After that, subtract the smallest value from the highest value in the set.

Question 5: Is grouped data continuous or discrete?

Answer: Usually, the continuous data is recorded as grouped data so it is usually represented by histograms or cumulative frequency graphs. Besides, sometimes also we record discrete data in groups to make calculations quicker.

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