Mean deviation is a measure of central tendency. We can calculate it from Arithmetic Mean, Median or Mode. It shows us how far are all the observations from the middle, on average? Each deviation is an absolute deviation as it is an absolute value which implies that we ignore the negative signs. Also, the deviations on both the sides of the Mean shall be equal. Let us start learning the definition of deviation first and then the mean deviation formula in detail.
What is Mean Deviation
Deviation in statistics is a measure that refers to the difference between the observed and expected values of a variable. In layman’s terms, a deviation is a distance from the centre point. Mean, median, and mode are all data set centre points. Similarly, the mean deviation is used to calculate the distance between the values in a data collection and the centre point.
Mean deviation is a statistical measure that computes the average deviation from the average value of a given data collection. The mean deviation can be calculated using various data series, such as – continuous data series, discrete data series and individual data series.
Mean Deviation Formula
The mean deviation is the mean of the absolute deviations of the observations or values from a suitable average. This suitable average may be the mean, median or mode. We also know it as the mean absolute deviation.
The basic formula to calculate mean deviation for a given data set is as follows:
where,
X = denotes each value in the data set
\(\overline{X}\) = denotes the mean value of the data set
N = total number of data values
| | = represents absolute value, i.e. it ignores the sign
How to calculate Mean Deviation?
Step 1 – Calculate the mean, median or mode value of the given data set.
Step 2 – Then we must find the absolute difference between each value in the data set with the mean, ignoring the signs.
Step 3 – We then sum up all the deviations.
Step 4 – Finally, we find the mean or average of those values found in Step 3. The result obtained is the mean deviation.
Let us look at a simple example to understand the working of the above steps. Suppose we have a dataset {2, 4, 8, 10} and we want to calculate the mean deviation about the mean.
Step 1 – We find the mean of the dataset i.e. (2+4+8+10)/4 = 6.
Step 2 – We then subtract each value in the dataset with the mean, get their absolute values i.e. |2-6| = 4, |4-6| = 2, |8-6| = 2, |10-6| = 4
Step 3 – And add them i.e. 4+2+2+4 = 12.
Step 4 – Finally, we divide this sum by the total number of values in the dataset (4) that will give us the mean deviation. The answer is 12/4 = 3.
Mean Deviation Types
- Individual Data Series – When the given data set is on an individual basis.
Age | 5 | 10 | 15 | 20 | 25 | 30 |
- Discrete Data Series – When the data set is given along with their frequencies.
Shoe Size | 7 | 8 | 9 | 10 |
Frequency | 5 | 10 | 7 | 13 |
- Continuous Data Series – When the data set is based on ranges along with their frequencies.
Age | 5-10 | 10-15 | 15-20 | 20-25 |
Frequency | 19 | 23 | 31 | 26 |
We shall now learn more about some important formulas, for example, the mean deviation formula for ungrouped data, ungrouped data as well as for an individual series or a continuous series, etc.
Mean Deviation Formula for Ungrouped Data
Ungrouped data is data that has not been sorted or categorized into groups and is still in its raw form. Generally, ungrouped data includes individual data series. The formula to calculate mean deviation for ungrouped data is as follows:
where,
x_{i} = i^{th} observation
\(\overline{X}\) = central point of the data (mean, median or mode)
n = number of observations
Mean Deviation Formula for Grouped Data
Grouped data is data that has been sorted and classified into groups. Continuous and discrete frequency distributions are used to group data.
- For Discrete Frequency Distribution – The formula to calculate mean deviation is as follows:
where,
x_{i} = specified individual observation
f_{i} = frequency of the occurrence of that observation
- For Continuous Frequency Distribution – The formula to calculate mean deviation is as follows:
where,
x_{i} = mid-value of the class intervals
f_{i} = frequency of repetition of x_{i}
Mean Deviation about Mean
The mean is calculated by taking the sum of all observations and dividing it by the total number of observations. Formulas for mean deviation about the mean are given below:
- For Individual Data –
- For Continuous/Discrete Data –
Mean Deviation about Median
The median is the value that separates the lower and upper halves of the data. The median is the number in the middle of a sorted, ascending or descending list of numbers. Formulas for mean deviation about the median are given below:
- For Individual Data –
where median (M) –
- For Discrete Data –
The median of discrete data is calculated as above
- For Continuous Data –
where,
c.f = cumulative frequency preceding the median class
l = lower value of the median class
f = frequency of the median class
h = length of the median class
Mean Deviation about Mode
The value that occurs the most frequently in a given data collection is defined as the mode. Formulas to calculate mean deviation about the mode is given below:
- For Individual Data –
where mode = most frequently occurring value in the data set
- For Discrete Data –
The mode of discrete data is calculated as above.
- For Continuous Data –
where,
l = lower value of the modal class
h = size of the modal class
f = frequency of the modal class
f_{1} = frequency of the class preceding the modal class
f_{2} = frequency of the class succeeding the modal class
Difference between Mean Deviation and Standard Deviation
Mean Deviation | Standard Deviation |
We use central points (mean, median, mode) to calculate the mean deviation. | To calculate the standard deviation we only use the mean. |
To calculate the mean deviation, we take the absolute value of the deviations. | We use the square of the deviations to calculate the standard deviation. |
It is less frequently used. | It is one of the most commonly used measures of variability and frequently used. |
When there are a greater number of outliers in the data, mean absolute deviation is employed. | When there are fewer outliers in the data, the standard deviation is employed. |
Merits of Mean Deviation
- Mean deviation is easy to understand and calculate.
- It gets least affected by extreme values (outliers).
- Mean Deviation is calculated by considering all the items in the data set.
- When compared to other statistical measures, it exhibits the fewest sample volatility.
- It is a useful comparison metric because it is based on deviations from the mean.
Demerits of Mean Deviation
- It is not strictly defined because it can be calculated in relation to the mean, median, and mode.
- Because we take the absolute value, we ignore both negative and positive indications. This can result in inaccuracies in the outcome.
- It is not a well-defined statistic because the mean deviation from different averages (mean, median, and mode) will differ.
- It cannot be further algebraically treated.
- The fluctuations in sampling have a significant impact on it.
Formula for the Co-efficient of Mean Deviation
- Co-efficient of Mean Deviation from Mean = \(\frac{M.D.}{ \overline{X}}\)
- Co-efficient of Mean Deviation from Median = \(\frac{M.D.}{M}\)
- The Co-efficient of Mean Deviation from Mode = \(\frac{M.D.}{Mode}\)
FAQs on Mean Deviation
Question 1. Calculate the mean deviation from the median and the co-efficient of mean deviation from the following data:
Marks of the students: 86, 25, 87, 65, 58, 45, 12, 71, 35.
Solution: Arrange the data in ascending order: 12, 25, 35, 45, 58, 65, 71, 86, 87.
Median = Value of the \(\frac{N+1}{2}^{th} term\)
= Value of the \(\frac{9+1}{2}^{th} term = 58\)
Calculation of mean deviation:
X | \(\left | X – M \right |\) |
12 | 46 |
25 | 33 |
35 | 23 |
45 | 13 |
58 | 0 |
65 | 7 |
71 | 13 |
86 | 28 |
87 | 29 |
N = 9 | \(\sum \left | X – M \right | = 460\) |
M.D. = \(\frac{\sum \left | X – M \right |}{N}\)
= \(\frac{460}{9}\)
= 51.11
Co-efficient of Mean Deviation from Median = \(\frac{M.D.}{M}\)
= \(\frac{51.11}{58}\)
= 0.881
Question 2. Calculate the mean deviation from mean for the following data.
x | 12 | 9 | 6 | 18 | 10 |
f | 7 | 3 | 8 | 1 | 2 |
Answer.
x | f | x.f | |x – µ| | f. |x – µ| |
12 | 7 | 84 | 2.619 | 18.33 |
9 | 3 | 27 | 0.381 | 1.143 |
6 | 8 | 48 | 3.381 | 27.048 |
18 | 1 | 18 | 8.619 | 8.619 |
10 | 2 | 20 | 0.619 | 1.238 |
Total | 21 | 197 | 56.378 |
We first find the Mean of the given dataset,
Finally, we substitute values in the mean deviation about mean formula,
Hence, the mean deviation about the mean is found to be 2.684
Question 3. Calculate the mean deviation for the following data.
Class Interval | 0 – 2 | 2 – 4 | 4 – 6 | 6 – 8 |
Frequency | 4 | 2 | 5 | 3 |
Answer.
Class Interval | Mid-point (x) | Frequency (f) | f.x | |x – µ| = |x – 4| | f. |x – µ| |
0 – 2 | 1 | 4 | 4 | 3 | 12 |
2 – 4 | 3 | 2 | 6 | 1 | 2 |
4 – 6 | 5 | 5 | 25 | 1 | 5 |
6 – 8 | 7 | 3 | 21 | 3 | 9 |
Total | 14 | 56 | 28 |
Finally, we substitute values in the mean deviation formula,
I get a different answer for first example.
I got Q1 as 20.5
median 23 and
Q3 26
Hi
Same
yes