Standard Deviation and Coefficient of Variation

Define standard deviation.

A Standard deviation is defined as the positive square root of the mean of the square of deviation from the arithmetic mean. It is also known as root mean square deviation. It is denoted by the Greek letter 'σ' (sigma).

What is coefficient of standard deviation? Also, write its formula.

The coefficient of standard deviation is the ratio of standard deviation (σ) to the mean (X̅).

Its formula is:
Coefficient of standard deviation = $\frac{σ}{\overline{X}}$

Define coefficient of variation. Also, write its formula.

If the coefficient of standard deviation is multiplied by 100, it is known as coefficient of variation. It is denoted by C.V.

Its formula is:
CV = $\frac{\text{standard deviation}}{\text{mean}}$ × 100%

It is purely a number and independent of unit.

What is the meaning a low coefficient of variation?

Lower the coefficient of variation greater will be the consistency or uniformity or more homogenous or more stable.

Write the difference between standard deviation and mean deviation.

Standard deviation (σ) is the positive square root of mean of the square of deviation from the arithmetic mean. It is also known as root mean square deviation.

Mean deviation (MD) is the average of the absolute values of the deviation of each item from mean, median or mode. It is also known as average deviation.

Write the difference between coefficient of standard deviation and the coefficient of variation.

Coefficient of standard deviation is the ratio of standard deviation (σ) to the mean (X̅) whereas coefficient of variation is the multiplication of the coefficient of standard deviation by 100.

Coefficient of standard deviation = $\frac{σ}{\overline{X}}$
Coefficient of variation = $\frac{σ}{\overline{X}}$ × 100%

In a continuous series, ∑𝑓(𝑚 - X̅)² = 484, N = 24 and X̅ = 25 find the standard deviation and its coefficient.

Here,
∑𝑓(𝑚 - X̅)² = 484
N = 24
X̅ = 25
Standard deviation (σ) = ?
Coefficient of σ = ?

Now, we have,
σ = $\sqrt{\frac{∑𝑓(𝑚 - \overline{X})^2}{N}}$ = $\sqrt{\frac{484}{24}}$ = 4.49

Coefficient of σ = $\frac{σ}{\overline{X}}$ = $\frac{4.49}{25}$ = 0.179

In a continuous series ∑fd = 0, ∑fd² = 848, N = 100, assumed mean (A) = 12 then find the standard deviation and its coefficient.

Here,
∑fd = 0
∑fd² = 848
N = 100
Assumed mean (A) = X̅ = 12
Standard deviation (σ) = ?
Coefficient of σ = ?

Now, we have,
σ = $\sqrt{\frac{\sum fd^2}{N} - (\frac{\sum fd}{N})^2}$ = $\sqrt{\frac{848}{100} - 0}$ = 2.91

Coefficient of σ = $\frac{σ}{\overline{X}}$ = $\frac{2.91}{12}$ = 0.24

If ∑fd'² = 125, ∑fd' = -4, N = 20, C = 10, find SD (σ).

Here,
∑fd'² = 125
∑fd' = -4
N = 20
C = 10
Standard deviation (σ) = ?

Now, we have, σ = $\sqrt{\frac{\sum fd'^2}{N} - (\frac{\sum fd'}{N})^2}$ × C or, σ = $\sqrt{\frac{125}{20} - \frac{16}{400}}$ × 10
∴ σ = 24.92