In this post, we will discuss about properties of Arithmetic mean with Examples.
Arithmetic Mean
The arithmetic mean, often simply referred to as the mean, is a statistical measure that represents the central value of a dataset. The arithmetic mean is calculated by summing all the values in the dataset and then dividing by the total number of observations in the data.
Table of Contents
The Sum of Deviations From the Mean is Zero
Property 1: The sum of deviations taken from the mean is always equal to zero. Mathematically $\sum\limits_{i=1}^n (x_i-\overline{x}) = 0$
Consider the ungrouped data case.
| Obs. No. | $X$ | $X_i-\overline{X}$ |
|---|---|---|
| 1 | 81 | -19 |
| 2 | 100 | 0 |
| 3 | 96 | -4 |
| 4 | 108 | 8 |
| 5 | 90 | -10 |
| 6 | 102 | 2 |
| 7 | 104 | 4 |
| 8 | 103 | 3 |
| 9 | 100 | 0 |
| 10 | 109 | 9 |
| 11 | 91 | -9 |
| 12 | 116 | 16 |
| Total | $\sum X_i = 1200$ | $\sum\limits_{i=1}^n (X_i-\overline{X})=0$ |
For grouped data $\overline{X} = \sum\limits_{i=1}^k f_i(X_i -\overline{X}) =0$, where for grouped data $\overline{X} =\frac{\sum\limits_{i=1}^n M_i f_i}{\sum\limits_{i=1}^n f_i}$. Suppose, we have the following grouped data
| Classes | $f$ | $M$ | $fM$ | $f_i(M_i – \overline{X})$ |
|---|---|---|---|---|
| 65 – 85 | 9 | 75 | 675 | $9\times (75 – 123) = -432$ |
| 85 – 105 | 10 | 95 | 950 | $10\times (95 – 123) = -280$ |
| 105 – 125 | 17 | 115 | 1955 | $17\times (115 – 123) = -136$ |
| 125 – 145 | 10 | 135 | 1350 | $10\times (135 – 123) = 120$ |
| 145 – 165 | 5 | 155 | 775 | $5\times (155 – 123) = 160$ |
| 165 – 185 | 4 | 175 | 700 | $4\times (175 – 123) = 208$ |
| 185 – 205 | 5 | 195 | 975 | $5\times (195 – 123) = 360$ |
| Total | $\Sigma f = n = 60$ | $\Sigma fM = 7380$ | $\sum\limits_{i=1}^k f_i(X_i -\overline{X}) =0$ |
Mean = $\overline{X} = \frac{\Sigma fM}{\Sigma f} = \frac{7380}{60} = 123$ .
The Combined Mean of Different Data Sets
Property 2: If there are different sets of data say $k$ then the combined mean/ average is
\begin{align*}
\overline{X}_c &= \frac{n_1 \overline{x}_1 + n_2\overline{x}_2 +\cdots + n_k \overline{x}_k }{n_1+n_2\cdots + n_k}\\
&=\frac{\Sigma x_1 + \Sigma x_2 + \cdots + \Sigma x_k}{n_1+n_2+\cdots + n_k}
\end{align*}
Suppose, we have data of $k$ groups.
| Obs. No. | $X_1$ | $X_2$ | $X_3$ | $X_4$ | $X_5$ |
|---|---|---|---|---|---|
| 1 | 81 | 40 | 92 | 107 | 113 |
| 2 | 100 | 30 | 95 | 110 | 94 |
| 3 | 96 | 22 | 99 | 114 | 93 |
| 4 | 108 | 51 | 94 | 109 | 119 |
| 5 | 90 | 101 | 116 | 105 | |
| 6 | 102 | 103 | 118 | ||
| 7 | 104 | 100 | 115 | ||
| 8 | 103 | 102 | |||
| 9 | 100 | 101 | |||
| 10 | 109 | ||||
| 11 | 91 | ||||
| 12 | 116 | ||||
| Sum | 1200 | 143 | 887 | 789 | 524 |
For \begin{align*}
\overline{X}_1 &= \frac{\sum\limits_{i=1}^n X_1}{n_1} = \frac{1200}{12} = 100\\
\overline{X}_2 &= \frac{\sum\limits_{i=1}^n X_2}{n_2} = \frac{143}{4} = 35.8\\
\overline{X}_3 &= \frac{\sum\limits_{i=1}^n X_3}{n_3} = \frac{887}{9} = 98.6\\
\overline{X}_4 &= \frac{\sum\limits_{i=1}^n X_4}{n_4} = \frac{789}{7} = 112.7\\
\overline{X}_5 &= \frac{\sum\limits_{i=1}^n X_5}{n_5} = \frac{524}{5} = 104.8\\
\overline{X}_c &= \frac{n_1\overline{X}_1 + n_2 \overline{X}_2 + \cdots + n_5 \overline{X}_5}{n_1+n_2+n_3+n_4+n_5}\\
&=\frac{12\times 100 + 4\times 35.8 + 9\times 98.6 + 7\times 112.7 + 5\times 104.8}{12+4+9+7+5} =\frac{3543.5}{37} = 95.7703
\end{align*}
For combined mean, not all the data set needs to be ungrouped or grouped. It may be possible that some data sets are ungrouped and some data sets are grouped.
Sum Squared Deviations from the Mean are Always Minimum
Property 3: The sum of the squared deviations of the observations from the arithmetic mean is minimum, which is less than the sum of the squared deviations of the observations from any other values. In other words, the sum of squared deviations from the mean is less than the sum of squared deviations from any other value. Mathematically,
For Ungrouped Data: $\Sigma (X_i – \overline{X})^2 < \Sigma (X_i – A)^2$
For Grouped Data: $\Sigma f(X_i – \overline{x})^2 < \Sigma f(M_i – A)^2$
where $A$ is any arbitrary value, also known as provisional mean. For this condition, $A$ is not equal to the arithmetic mean.
Note that the difference between the sum of deviations and the sum of squared deviations is that in the sum of deviations we take the difference (subtract) of each observation from the mean and then sum all the differences. In the sum of squared deviations, we take the difference of each observation from the mean, then take the square of all the differences, and then sum all the resultant values at the end.
From the above calculations, it can observed that $\Sigma (X_i – \overline{X})^2 < \Sigma (X_i – 90)^2 < \Sigma (X_i – 99)^2$.
No Resistant to Outliers
Property 4: The arithmetic mean is not resistant to outliers. It means that the arithmetic mean can be misleading if there are extreme values in the data.
Arithmetic Mean is Sensitive to Outliers
Property 5: The arithmetic mean is sensitive to extreme values (outliers) in the data. If there are a few very large or very small values, they can significantly influence the mean.
The Affect of Change in Scale and Origin
Property 6: If a constant value is added or subtracted from each data point, the mean will be changed by the same amount.
Similarly, if a constant value is multiplied or divided by each data point, the mean will be multiplied or divided by the same constant.
Unique Value
Property 7: For any given dataset, there is only one unique arithmetic mean.
In summary, the arithmetic mean is a widely used statistical measure (a measure of central tendency) that provides a central value for a dataset.
However, it is important to be aware of the properties of arithmetic mean and its limitations, especially when dealing with data containing outliers.
FAQs about Arithmetic Mean Properties
- Explain how the sum of deviation from the mean is zero.
- What is meant by unique arithmetic mean for a data set?
- What is arithmetic mean?
- How combined mean of different data sets can be computed, explain.
- Elaborate Sum of Squared Deviation is minimum?
- What is the impact of outliers on arithmetic mean?
- How does a change of scale and origin change the arithmetic mean?
