Measure of Dispersion - Statistics | Mathematics (Maths) Class 11 - Commerce PDF Download

Measure of Dispersion

The degree to which numerical data tend to spread about an average value is called the dispersion of the data. The four measure of dispersion are

1. Range
2. Mean deviation
3. Standard deviation

4. Variance

Range

The difference between the highest and the lowest element of a data called its range.

i.e., Range = X_{max – Xmin}

∴ The coefficient of range = X_{max – Xmin / Xmax + Xmin}

It is widely used in statistical series relating to quality control in production.

(i) Inter quartile range = Q₃ — Q₁

(ii) Semi-inter quartile range (Quartile deviation)

∴ Q D = Q₃ — Q₁ / 2

and coefficient of quartile deviation = Q₃ — Q₁ / Q₃ + Q₁

(iii) QD = 2 / 3 SD

Mean Deviation (MD)

The arithmetic mean of the absolute deviations of the values of the variable from a measure of their Average (mean, median, mode) is called Mean Deviation (MD). It is denoted by δ.

(i) For simple (discrete) distribution

δ = Σ |x – z| / n

where, n = number of terms, z = A or M_d or M_o

(ii) For unclassified frequency distribution

δ = Σ f |x – z| / Σ f

(iii) For classified distribution

δ = Σ f |x – z| / Σ f

Here, x is for class mark of the interval.

(iv) MD = 4 / 5 SD

(v) Average (Mean or Median or Mode) = Mean deviation from the average / Average

Note The mean deviation is the least when measured from the median.

Coefficient of Mean Deviation

It is the ratio of MD and the mean from which the deviation is measured. Thus, the coefficient of MD

= δ A / A or δ M _d / M _d or δ M _o / M _o

Standard Deviation (σ)

Standard deviation is the square root of the arithmetic mean of the squares of deviations of the terms from their AM and it is denoted by σ.

The square of standard deviation is called the variance and it is denoted by the symbol σ².

(i) For simple (discrete) distribution

(ii) For frequency distribution

(iii) For classified data

Here, x is class mark of the interval.

Shortcut Method for SD σ =

where, d = x — A’ and A’ = assumed mean

Standard Deviation of the Combined Series

If n₁, n₂ are the sizes, X₁, X₂ are the means and σ₁, σ₂ are the standard deviation of the series, then the standard deviation of the combined series is

where,

Variance:

Standard Deviation: The standard deviation is the positive square root of the variance. It provides a measure of the dispersion or spread of a set of data points. It is denoted by $\sigma \backslash sigma$ or S.D.

Formulas

Variance:

Properties of Variance

1. Addition of a Constant:  var(x_i+λ) = var(x_i)  Adding a constant $\lambda \backslash lambda$λ to each data point does not change the variance.

2. Multiplication by a Constant:  $var\backslash text\{var\}(\backslash lambda\; x\_i)\; =\; \backslash lambda^2\; \backslash text\{var\}(x\_i)$var(λx_i)=λ²var(x_i)  Multiplying each data point by a constant $\lambda \backslash lambda$λ scales the variance by $\lambda 2\backslash lambda^2$λ2.

3. Linear Transformation:  $var$var(ax_i+b)=a²var(x_i) For a linear transformation where $x\_i$x_i is multiplied by a constant $aa$a and then added to a constant $bb$, the variance is scaled by $a^2$a². The addition of the constant b does not affect the variance.

Effects of Average and Dispersion on Change of origin and Scale