IIT JEE Main Maths -Unit 13- Standard deviation, variance, mean deviation- Study Notes-New Syllabus

Measures of Dispersion

Measures of dispersion tell us how spread out the values of a dataset are around the average. The three most important measures in JEE are:

• Mean Deviation (M.D.)
• Variance
• Standard Deviation (S.D.)

 Mean Deviation (M.D.)

Mean deviation is the average of absolute deviations from mean or median.

(a) Ungrouped Data

\( \text{MD about mean} = \dfrac{\sum |x_i – \bar{x}|}{n} \)

\( \text{MD about median} = \dfrac{\sum |x_i – M|}{n} \)

(b) Grouped Data

\( \text{MD about mean} = \dfrac{\sum f_i |x_i – \bar{x}|}{\sum f_i} \)

\( \text{MD about median} = \dfrac{\sum f_i |x_i – M|}{\sum f_i} \)

Mean deviation about the mean is smaller than about the median for symmetrical distributions.

Variance

Variance measures the average squared deviation from the mean.

(a) Ungrouped Data

\( \sigma^2 = \dfrac{\sum (x_i – \bar{x})^2}{n} \)

Shortcut formula:

\( \sigma^2 = \dfrac{\sum x_i^2}{n} – \bar{x}^2 \)

(b) Grouped Data

\( \sigma^2 = \dfrac{\sum f_i (x_i – \bar{x})^2}{\sum f_i} \)

Shortcut:

\( \sigma^2 = \dfrac{\sum f_i x_i^2}{\sum f_i} – \bar{x}^2 \)

Standard Deviation (S.D.)

S.D. is the positive square root of variance.

\( \sigma = \sqrt{\sigma^2} \)

S.D. is the most important measure of dispersion for JEE.

 Coding Method (Shortcut Method) Used for large values in grouped data.

If

\( u_i = \dfrac{x_i – a}{h} \)

Then

\( \bar{x} = a + h\dfrac{\sum f_iu_i}{\sum f_i} \)

\( \sigma = h\sqrt{\dfrac{\sum f_i u_i^2}{\sum f_i} – \left(\dfrac{\sum f_i u_i}{\sum f_i}\right)^2} \)

Comparison of Dispersions

• \( \text{S.D.} \ge \text{Mean Deviation} \ge \text{Quartile Deviation} \)
• S.D. is widely used because of algebraic convenience.