Standard Deviation is a measure of dispersion that tells us how far the values of a series tend to deviate from their arithmetic mean.

In simple words, standard deviation helps us understand how much the data values are spread out around their average value.

• Small standard deviation The values are close to the mean.
• Large standard deviation The values are spread farther away from the mean.
• Zero standard deviation All the values are equal.

Standard deviation is denoted by the Greek letter σ (sigma) for a population. It is calculated by taking the square root of the arithmetic mean of the squared deviations from the arithmetic mean.

Standard deviation of individual data is a statistical method used to measure the dispersion or spread of individual observations around their arithmetic mean.

In an individual series, data values are given separately, and each observation is considered individually. There are no frequencies assigned to the values.

Example of individual data

The marks obtained by five students are:

Here, each value represents an individual observation. We calculate standard deviation to determine how far these marks are spread around their average.

In this method, to find the standard deviation, the actual mean (X̄) must be calculated. If the mean is a decimal value, the calculation may become somewhat difficult.

The following formula is used to calculate the standard deviation using this method:

• d = X − X̄ : the difference between each item of the data and the mean
• N = total number of items
• X̄ = mean value of the data

Steps for calculating standard deviation by the Actual Mean Method

1. Find the mean using the formula: X̄ = ∑XN
2. Find the deviation of each given value X from the mean: d = X − X̄.
3. Find the square of each deviation d² and calculate the total ∑d².
4. Finally, calculate the standard deviation using the formula: S.D. = ∑d²N

In this method, the actual mean and the assumed mean are not required to calculate standard deviation. The sum of all given values X is calculated, and each value is squared to find ∑X².

Then the standard deviation is calculated using the following formula:

• X = the value of any item
• N = total number of items
• ∑X = sum of all the values
• ∑X² = sum of the squares of all the values

Steps

1. Find ∑X by adding all the values.
2. Square every value and find ∑X².
3. Count the number of items N.
4. Put the values in the formula and simplify.

It is difficult and time-consuming to calculate the standard deviation using the actual mean when the values are large or complicated. In such a situation, one value from the data is taken as the assumed mean, and the standard deviation is calculated based on it.

To find the standard deviation by this method, the given data is first written in ascending order, and a central value is taken as the assumed mean (A).

The formula used is:

• d = X − A = the deviation of each item from the assumed mean
• A = assumed mean
• N = total number of items
• X = value of any item

Steps for calculating standard deviation by the Assumed Mean Method

1. Choose a central value from the given data as the assumed mean A.
2. Find the deviation of each value from the assumed mean: d = X − A.
3. Find the sum of deviations ∑d.
4. Calculate the square of deviations d² and find ∑d².
5. Apply the formula of standard deviation: S.D. = ∑d²N − (∑dN)²

Standard deviation in a discrete frequency distribution is a statistical method used to measure the dispersion or spread of observations around their arithmetic mean when each value has a corresponding frequency.

In a discrete frequency distribution, data values are presented separately along with their frequencies. The frequency indicates how many times a particular value occurs.

For example, if the value 10 has a frequency of 3, it means that the value 10 occurs three times in the dataset.

Standard deviation helps us understand whether the values are concentrated near the mean or spread far away from it.

Example of a discrete frequency distribution

Here:

• X represents the observations.
• f represents the frequency of each observation.
• ∑f represents the total frequency.

Since the values are presented with their frequencies, we use a formula that considers both X and f.

The Actual Mean Method is a method of calculating standard deviation in which we first calculate the actual arithmetic mean of the given data and then find the deviations of each observation from that mean.

In this method, we use the actual mean X̄ to calculate the deviations.

The formula for standard deviation is:

Where:

• σ = standard deviation
• f = frequency
• d = X − X̄ = deviation from the actual mean
• X = value of the variable
• X̄ = actual arithmetic mean, X̄ = ∑fXN
• N = ∑f = total frequency

The Direct Method is a method of calculating standard deviation in which we use the original values of the variable and their corresponding frequencies to calculate standard deviation.

In this method, we do not need to calculate deviations from the mean separately. Instead, we calculate the square of each value (X²), multiply it by its frequency (fX²), and apply the direct formula.

The Direct Method is particularly useful when the values of a discrete frequency distribution are small and easy to calculate.

Formula

Where:

• σ = standard deviation
• X = value of the variable
• f = frequency
• X² = square of the value
• fX = product of frequency and value
• fX² = product of frequency and squared value
• N = ∑f = total frequency

The Assumed Mean Method is a method of calculating standard deviation in which we select a convenient value from the given data as the assumed mean and calculate the deviations of the observations from that value.

This method is useful when the actual mean is difficult to calculate or when the values are large.

Instead of calculating deviations from the actual mean, we calculate deviations from the assumed mean, which simplifies the calculations.

Formula

Where:

• σ = standard deviation
• f = frequency
• d = X − A = deviation from the assumed mean
• X = value of the variable
• A = assumed mean
• N = ∑f = total frequency
• ∑fd = sum of frequency multiplied by deviation
• ∑fd² = sum of frequency multiplied by squared deviation