Find the mean and standard deviation using short-cut method.
\(x_i\) | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 |
\(f_i\) | 2 | 1 | 12 | 29 | 25 | 12 | 10 | 4 | 5 |
The data is obtained in tabular form as follows.
\(x_i\) | \(f_i\) | \(fx_i=\frac{x_i-64}{1}\) | \(y_1^2\) | \(f_iy_i\) | \(f_iy_1^2\) |
60 | 2 | -4 | 16 | -8 | 32 |
61 | 1 | -3 | 9 | -3 | 9 |
62 | 12 | -2 | 4 | -24 | 48 |
63 | 29 | -1 | 1 | -29 | 29 |
64 | 25 | 0 | 0 | 0 | 0 |
65 | 12 | 1 | 1 | 12 | 12 |
66 | 10 | 2 | 4 | 20 | 40 |
67 | 4 | 3 | 9 | 12 | 36 |
68 | 5 | 4 | 16 | 20 | 80 |
100 | 220 | 0 | 286 |
Mean, \(\bar{x}=A\frac{\sum_{i=1}^9f_ix_i}{n}×h=64+\frac{0}{100}×1=64+0=64\)
Variance, (σ2) = \(\frac{h^2}{N^2}[N\sum_{i=1}^9f_iy_i^2-(\sum_{i=1}^9f_iy_i)^2]\)
\(=\frac{1}{100^2}[100×286-0]\)
\(=2.86\)
\(∴\,standard\,deviation\,(σ)=√2.86=1.69\)
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According to layman’s words, the variance is a measure of how far a set of data are dispersed out from their mean or average value. It is denoted as ‘σ2’.
Read More: Difference Between Variance and Standard Deviation
The spread of statistical data is measured by the standard deviation. Distribution measures the deviation of data from its mean or average position. The degree of dispersion is computed by the method of estimating the deviation of data points. It is denoted by the symbol, ‘σ’.
1. Population Standard Deviation
2. Sample Standard Deviation