The positive part of the square root of the variance is the standard deviation. One of the most basic approaches to statistical analysis is the standard deviation. The acronym for standard deviation is SD, represented by the letter sigma, which is ‘σ’. Standard variation indicates how far a value has varied from the mean value. If the standard deviation is low, the values are close to the mean, whereas a large standard deviation indicates that the values are significantly different from the mean.

In this article, let’s look at how to determine the standard deviation of grouped and ungrouped data and the random variable’s standard deviation.

**What is Standard Deviation?**

Standard deviation or SD is a concept of descriptive statistics. The standard deviation measures how dispersed or scattered the data points are about the mean. It describes how the values are distributed over the data sample and is an estimate of the data points’ deviation from the mean. The square root of the variance is the standard deviation of multiple parameters like the sample, statistical population, random variable, data collection, and probability distribution.

Let us suppose that we have n number of observations which are:

(a_{1}, a_{2}, a_{3}, ………, a_{n})

The mean deviation, in this case, will be formulated as

i= 1n (ai – a’)^{2}.

This formula is an extremely reliable and reasonable indicator of scattering or dispersion. We say this because when we determine the sum of squares of deviations obtained from the mean, it gives error results, and we cannot consider it a proper measure of dispersion. Using standard deviation for measurement tells us that if the observations are close to the mean when the average of the squared difference from the mean is a smaller value. This is also known as a lower degree of dispersion or scattering. If the value of the squared difference is greater than the value of the mean, then the degree of dispersion is high. It indicates that the value is greater than the mean. Let us take an example to grasp this concept.

**Example:** See the data points given below

4, 5, 6, 8

and

2, 3, 8, 11

**Solution: **When we calculate the mean of 4, 5, 6, 8 we get the answer as

(4 + 5 + 6 + 8)/4 = 6

Similarly,

(2 + 3 + 8 + 11)/4 = 6.

We see that both sets of observations have the same mean value. But when we look closely, the numbers in the second part are much farther apart from each other than the numbers in set 1. This is why standard deviation is required.

Add to your knowledge: The square of standard deviation is called the variance. It is denoted by σ^{2} and is also a good parameter to determine the accurate measure of the deviation.

**Math Involved: Standard Deviation Formula**

The main application of standard deviation is to measure the dispersion of statistical data. Since it deals with data, it involves mathematics. Determining the divergence of data points is used to calculate the degree of dispersion. On the other hand, the standard deviation is the range of data values that are close to the mean. Here are two standard deviation formulas. They are known as population standard deviation and sample standard deviation. Let us learn about them one at a time.

Population Standard Deviation→

σ_{p} = 1Ni = 1N (a_{i} – η)^{2}

This is the formula for population standard deviation. Here,

σ_{p} = the standard deviation for the population

N = the total number of observing the population

η = assumed mean

a_{i} = the desired value of the number

Similarly, we have the formula of Sample Standard deviation→

σ_{s} = 1ni = 1n (a_{i} – a’)^{2}

In the formula

σ_{s} = the standard deviation for samples

N = the total number of samples observed

a’ = mean of the samples

a_{i} = the desired value of the number

**Steps to Calculate Standard Deviation**

Follow the steps mentioned below to calculate the standard deviation of the ‘n’ number of terms.

**Step 1: **You need to calculate the arithmetic means of all the observations

**Step 2: **Next, determine the square difference between the data value and mean. ( k = Data value – mean )2.

**Step 3: **Now you have to find the average of all the squared differences. This is the variance of the observation. (k divided by the number of operations)

**Step 4:** Square root the value of the variance and you are left with the standard variation.

(Variation )

**Steps to Calculate Sample Standard Deviation**

In the above section, we learned how to calculate the standard deviation. From this section onward, we will study the method to calculate sample standard deviation. Look at the steps below:

**Step 1:** First, calculate the mean of the entire data, which is a’ in the formula.

**Step 2:** Take the mean of each data point and subtract it. Deviations are the term for these differences. Negative deviations apply to data points below the mean, while positive deviations apply to data points above.

**Step 3:** To make each deviation positive, square it.

**Step 4: **Add or sum the squared deviations.

**Step 5:** Subtract one from the total to get the number of observations in the sample. The variance is the name given to the outcome.

**Step 6:** The standard deviation is calculated by taking the square root of the variance.

Let us look at an example to understand these steps easily.

**Example: **You are given 4 observations 2, 5, 6, 4. Find the sample standard deviation for these observations.

**Solution: **Let us look at this example with the steps mentioned above.

- Finding the value of a’ or the mean of all the observations.

(2 + 5 + 6 + 4)/4 = 4. Thus the sample of the mean is 4.

- Now we subtract each observation from the mean.

Observation | Result |

2 – 4 | -2 |

5 – 4 | 1 |

6 – 4 | 2 |

4 – 4 | 0 |

- Now we will make all the results into positive values by squaring them.

Observation | Result | Square of the result |

2 – 4 | -2 | (-2)^{2} = 4 |

5 – 4 | 1 | (1)^{2} = 1 |

6 – 4 | 2 | (2)^{2} = 4 |

4 – 4 | 0 | (0)^{2} = 0 |

- Next, we will add all the numbers from the last column, i.e., the result column.

4 + 1 + 4 + 0 = 9

- Divide the result with the number one less than the observation. Here we have 4 observations hence we will divide 9 by (4 – 1 = 3).

9/3 = 3. This is the variance of the observation.

- Just square the final result, that is the variance 3 to get the standard deviation.

3 = 1.732.

Hence the sample standard deviation for the observations 2, 5, 6, 4 = 1.732.

**Word Problem Example:** On the ground, 4 students kick the football at different meters. The data of individual kicks are given below. Find the sample standard deviation of the distance.

8, 6, 7, 3

**Solution:** Steps are already mentioned in the above example. We shall directly solve this question.

- (8 + 6 + 7 + 3)/4 = 6. Thus the sample of the mean is 6.
- Now we subtract each observation from the mean.

Observation | Result |

8 – 6 | 2 |

6 – 6 | 0 |

7 – 6 | 1 |

3 – 6 | -3 |

- Squaring the result.

Observation | Result | Square of the result |

8 – 6 | 2 | (2)^{2} = 4 |

6 – 6 | 0 | (0)^{2} = 0 |

7 – 6 | 1 | (1)^{2} = 1 |

3 – 6 | -3 | (3)^{2} = 9 |

- 4 + 0 + 1 + 9 = 14
- 14/3 = 4.66. This is the variance of the observation.
- Squaring the final result →

4.66 = 2.15.

Hence the sample standard deviation for the observations 8, 6, 7, 3 = 2.15.

**Standard Deviation of Probability Distribution**

When dealing with probability, you have to deal with numerous trials. In cases when the difference between probability in theories and relative frequency of the observation starts to come closer to one another, researchers can easily predict the result or mean. This means it is then expected to have a value denoted by the symbol 𝜇, for sample standard deviation of probability distribution note down these important points.

- When the distribution is zero, the standard deviation is one, and the mean of the observations is zero.
- In the case of binomial experiments, the probability of the event being successful is a random variable. The standard deviation given for probability is 𝜎= √npq. Here n = number of trials, p = success probability, and 1-p =q, which is the probability of failure. You can also write 𝜎= √uq, where u = np.
- The standard deviation in cases of Poisson distribution is formulated as 𝜎= √λt. Here λ is the average number of successes in time t.