Ungrouped data is the type of distribution in which the data is individually given in a raw form. For example, the scores of a batsman in last 5 matches are given as 45,34,2,77 and 80. Deduction of range and mean deviation from this data will help us to conclude his form and performance. Here, we will be studying methods to calculate range and mean deviation for individual series distribution.
Range and Mean Deviation
The range is defined simply as the difference between the maximum and minimum value in the distribution. Mean deviation is defined mathematically as the ratio of the summation of absolute values of dispersion to the number of observations.
In simple words, mean deviation represents the dispersion of all the data items in the series relative to the measure of central tendency, taken as median or mean in our course. Note that mean deviation about mode can also be calculated.
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Range of Ungrouped Data
We know now that range is the difference between the maximum and minimum value. Hence for ungrouped data, we arrange the series in ascending or descending order. This helps us to select the highest and lowest values in the distribution. Henceforth, we simply subtract the minimum value from the maximum value.
For example – The marks of a student in 5 tests of the chapter statistics are(out of 20)- 11, 14, 16, 13 and 18. Arranging them in ascending order- 18, 16, 14, 13 and 11. The range of the data is given as- 18-11=7.
Range= Maximum value – Minimum value
Mean Deviation for Ungrouped Data
As mentioned earlier, mean deviation measures the dispersion of data about a measure of central tendency. This measure of central tendency is generally median or mean. Let’s start with recalling how to calculate mean and median for individual distribution series.
Calculating Mean and Median
For calculation of median, we first arrange the data in ascending or descending order(generally ascending order). Further, we count the number of observations which is denoted by n. Now depending on whether n is even or odd, the further calculation is bifurcated as:
- If n is an odd number then the value of (n+1)/2th item is the median.
- If n is an even number then median is given as: [ value of (n+1)/2th item + value of (n/2 +1)th item]÷2
Mean is simply calculated as the ration of summation of observations to the number of observations.
Mean= Sum of observations/number of observations
Calculating Mean Deviation
After calculating mean and median we move forward to mean deviation. The value relative to which mean deviation is calculated is denoted by ‘a’. Further deviations for the data members are calculated as the modulus(absolute value) of the difference of ‘a’ from the data member. Deviation of a value X is given as |X – a|.
Lastly, the summation of these deviations for all data members is divided by the number of observations denoted by n.
Mean Deviation=[ ∑|X – a|]÷n
Here, ∑|X-a| = The summation of the deviations for values from ‘a’
n= The number of observations
Solved Example for You
Question 1: Find the mean deviation about median for the following:
Answer: Number of observations n = 7
As n is odd median is calculated as = value of (n+1)/2 th item= 8/2 th= 4th item= 8
Hence mean deviation= 15/7= 2.14
Question 2: State what is range deviation?
Answer: Awe can define the range as a single number that represents the spread of data. Simply, the range is the subtraction of the lowest score from the highest score. Moreover, for finding range you need scores that have some variability. Also, it can be easily understood.
Question 3: What is the formula of range with example?
Answer: Finding range is not difficult you just need to find the difference between the largest value of the set and the smallest value of the set. However, the formula of the range is = maximum value – minimum value. E.g. in the set data of 2, 4, 6, 8, 10, 12, 14, has the maximum value 14 and minimum value 2 so the range is 14-2 =12.
Question 4: State the range of a graph?
Answer: The domain of range refers to the set of possible input values. Moreover, the domain of a graph consists of all the input values shown on the x-axis and the range is the set of possible output values that are on the y-axis.
Question 5: Why is the range important?
Answer: It is important because it is the most obvious measure of dispersion and is the difference between the lowest and highest values in a dataset.