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Variance and Standard Deviation

 

In statistics, standard deviation and variance are the two most used concepts. This is because both need a mean of data for their calculations. Therefore, both variance and standard deviation are represented in the units of the mean. However, the standard deviation unit is the same as that of the mean, whereas variance has squared units as that of the mean.

Variance

Whenever we need to show the diversity of a data or its range, we use variance. Variance is the numerical representation of the range of data from its mean or average value. It is represented by 𝜎².

Since the variance is a square of any number, it will always be a positive value. It can either be positive or zero. For example, if the mean of any data is coming out -10. Then its variance will be (-10)² = 100. Hence, we will always get a positive value.

Variance formula

Population formula:

image

Here,

σ² = Population variance

N = Number of observations in population

Xᵢ = iᵗʰ observation in the population

μ = Population mean Sample variance formula:

image1

Here,

s² = Sample variance

n = Number of observations in sample

Xᵢ = iᵗʰ observation in the sample

X = Sample mean

Standard deviation

Whenever we need to measure the dispersion of any data, we use standard deviation. It is a measure to showcase the spread of the number of observations in data. It is represented by 𝜎. Standard deviation is the square root of variance. Therefore, it is known as the root-mean-square deviation.

Standard deviation cannot be negative. It will always be positive or zero. If the data is grouped with similar values, then the standard deviation will be nearer to zero. Likewise, if the data is widely spread, the value of standard deviation will also be more.

Standard deviation formula

Population formula:

The population standard deviation formula is given as:

image

Here,

σ = Population standard deviation Sample standard deviation formula:

image1

Here,

s = Sample standard deviation

Difference between variance and standard deviation

Variance Standard deviation
It is defined as the numerical value to describe the range or diversity of mean or average value of a data. It is a measure to showcase how far the data is. It denotes the dispersion of the data.
It is the square of deviation of the data. It is the root-mean-square of the data.
It is expressed in squared units. It is expressed in the same units as that of the mean.
It is denoted by 𝜎². It is denoted by 𝜎.
It is mostly used to indicate the individual spread in a group. It is used to indicate the number of observations in the spread of a data.

Example: Find the variance and standard deviation of the possibilities of a rolling of a die.

Solution:

The sample space of rolling of a die is {1, 2, 3, 4, 5, 6}.

The mean of the sample space = x̅ = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5

We know, the standard deviation formula is given by:

σ² = Σ (xi – x̅)² / n

σ² = ⅙ (6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25)

σ² = 2.917

Therefore, the standard deviation, σ = √2.917 = 1.708

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