JEE Mathematics Statistics Notes Set 02

Measures of Central Tendency

Arithmetic mean: \(\bar x=\frac{\sum x_i}{n}\) or \(\frac{\sum f_ix_i}{\sum f_i}\); step-deviation form \(\bar x=A+h\frac{\sum f_iu_i}{\sum f_i}\) with \(u=\frac{x-A}{h}\). The combined mean of two groups is \(\frac{n_1\bar x_1+n_2\bar x_2}{n_1+n_2}\), always between the two means. Sum of deviations from the mean is zero, and the mean changes with both change of origin and scale.

Geometric mean: \(G=(x_1x_2\cdots x_n)^{1/n}\), with \(\log G\) equal to the mean of the logarithms. Harmonic mean: \(H=\frac{n}{\sum\frac1{x_i}}\). For positive unequal numbers \(A>G>H\), and for two numbers \(G^2=AH\).

Median: the middle value of the ordered data; for grouped data \(l+\frac{\frac N2-C}{f}h\). Mode: most frequent value; for grouped data \(l+\frac{f_1-f_0}{2f_1-f_0-f_2}h\). For moderately skewed data, Mean \(-\) Mode \(=3(\)Mean \(-\) Median\()\).

Measures of Dispersion

Range = largest \(-\) smallest. Mean deviation about a: \(\frac{\sum|x_i-a|}{n}\), least about the median. Variance: \(\sigma^2=\frac{\sum x_i^2}{n}-\bar x^2\), and standard deviation \(\sigma=\sqrt{\sigma^2}\). For the first n natural numbers \(\sigma^2=\frac{n^2-1}{12}\).

Change of origin does not change the variance; multiplying every value by k multiplies the standard deviation by \(|k|\). Combined variance of two groups: \(\frac{n_1(\sigma_1^2+d_1^2)+n_2(\sigma_2^2+d_2^2)}{n_1+n_2}\), where \(d_i=\bar x_i-\bar x\). Coefficient of variation: \(\frac{\sigma}{\bar x}\times100\); the series with the larger CV is more variable.

Worked Examples

Example 1. Mode of the classes 40-50 (frequency 28) with neighbours 12 and 20: \(40+\frac{28-12}{56-12-20}\times10=46.67\).

Example 2. Mean 20 and median 21 give mode \(=3\times21-2\times20=23\).

Example 3. Marks of 50 students: Mathematics (mean 42, SD 12), Physics (32, 15), Chemistry (40.9, 20) have CV 28.57, 46.87 and 48.89, so Chemistry is the most variable and Mathematics the least.

Quick Tips

For corrected means, adjust the total, not the mean. Use the coefficient of variation to compare two series of different units or means.