Sample Standard Deviation Calculator

comma separated input values
Standard Deviation (sX)  =  3.0277
Variance (sX2)  =  9.1667

Standard Deviation - work with steps

Objective :
Find the standard deviation of sample
`x_i`: 8, 5, 2, 4, 10, 1, 7, 3, 6, 9

Input Data :
`x_i`: 8, 5, 2, 4, 10, 1, 7, 3, 6, 9

The number of values in a sample `n = 10`

Formula :

Solution :
`x_i``x_i - \bar{X}``(x_i - \bar{X})^2`
`\sum x_i ``= 55``\sum(x_i - \bar{X})^2`` = 82.5`
`\bar{X} = 55/10` `= 5.5`
`s_X=\sqrt{\frac{82.5}{10 - 1}`
Standard Deviation (`s_X`) = 3.0277
Variance (`s_X^2`) = 9.1667

Standard deviation calculator calculates the sample standard deviation from a sample `X : x_1, x_2, . . . , x_n`, using simple method. It’s an online Statistics and Probability tool requires a data set (set of real numbers or valuables). The result will describe the spread of dataset, i.e. how widely it is distributed about the sample mean.
It is necessary to follow the next steps:

  1. Enter a sample (observed values) in the box. These values must be real numbers or variables and may be separated by commas. The values can be copied from a text document or a spreadsheet.
  2. Press the "GENERATE WORK" button to make the computation.
  3. Standard deviation calculator will give the sample standard deviation of the sample `X : x_1,x_2,...,x_n.`
Input: A list of real numbers or a list of variables separated by comma
Output: A real number or a variable

standard deviation calculator gives us the stepwise procedure and insight into every step of calculation. Before the final result of sample standard deviation is derived, it calculates the arithmetic mean of a sample. This magnitude is called the sample mean and it is denoted by `\bar{X}`. The sample standard deviation calculator also calculates square of the sample standard deviation and get magnitude known as variance. These values of the sample mean and the variance can be of benefit for further solving of problems and applications.

Standard Deviation Formulas:
The sample mean
$$\bar{X}=\frac{1}{n}\sum_{i=1}^{i=n}x_i=\frac{x_1+\ \ldots+x_n}{n}$$
The sample standard deviation
sample standard deviation formula

What is Standard Deviation?

The standard deviation often abbreviated to SD, denoted by `s_X`, is an important magnitude of spread that is based on the mean. The mean can be interpreted as balancing point of a sample. The SD expresses how much the members of a sample differ from the mean value of the sample.

The difference between the each value of a sample and the mean is called the deviation. For instance, `x_1 − \bar{X}` is the deviation. The deviation of a value that is less than the mean is negative, since the deviation of a value that is greater than the mean is positive. If we add all the deviations, their sum will be 0. For this reason, we calculate the sum of the squared deviations. The SSD summarize all of the squared deviations.

Normally, for finding a mean of a sample, we divide by the number of members of a sample `n`. In the formula of standard deviation, instead of dividing by `n`, we divide by `(n − 1)`. In many experiments, dividing by `(n−1)` is often being used instead of dividing by entire population data `n` as because getting the entire population data for every experiments is practically not possible at all. This is called the sampling error. Therefore, it’s popularly known as sample standard deviation (often abbreviated as SSD) due to its usage of partial data samples of a population data. The SSD is always a positive number. So, when data set are negative or when the sample mean is `0`, then the SSD is greater than mean.

How to Calculate Standard Deviation?

By help of the standard deviation calculator, we can easily calculate the sample mean and standard deviation. The following is the step-by-step procedure for how to calculate stadard deviation manually, using the easiest tabular method.

  1. Find the number of samples to be included in the statistical experiment
  2. Find the mean
  3. Find the deviation between each member of sample and mean
  4. Find the square value of each difference
  5. Find the sum of square value of each difference
  6. Dividing the sum by n-1 is the variance
  7. The square root of variance is standard deviation

In many cases, we can calculate the sample standard deviation value by using the tabular (easy) method, especially for small calculations. But, if we use a large set of data for calculation or we want to get an accurate result, then we should use the standard deviation calculator.

The standard deviation work with steps shows the complete step-by-step calculation for finding the standard deviation and variance of a given sample of numbers `X : 5, 6, 8, 10`. For any other sample, just supply the list of numbers or variables and click on the "GENERATE WORK" button. The grade school students may use this standard deviation calculator to generate the work, verify the results derived by hand or do their homework problems efficiently.

Real World Problems Using Standard Deviation

The sample standard deviation can be useful in definition of some statistics, for example sample correlation coefficient. The sample correlation coefficient between samples `X and Y` is defined by the following formula

`\rho_{XY}=\frac{1}{n-1}\sum_{i-1}^{n}\frac{(x_i-\bar{X})(y_i-\bar{Y})}{s_X s_Y}`

where `\bar{X} and \bar{Y}` are the sample means and `s_X` and `s_Y` are sample standard deviations of samples `X and Y` respectively.

In statistical conclusions, we are interested to know whether the given sample comes from a population. We can use Z-Test to interference the population mean from the sample mean. In some cases, we do not have data of the whole population. Without the population standard deviation, we use Students t-Test to interference the population mean from sample mean.

There are many areas where the SSD may be applied in real life. One of them is pharmacy. For example, if a medicine lowers the blood pressures to patients with high blood pressure standard deviation has been applied to examine are there significant lowering of blood pressures. In finance, SSD of price data can be used as a measure of volatility. In manufacturing, it is used to estimate a quality of control. In polls, it measures the level of confidence of results.

Standard Deviation Practice Problems

Practice Problem 1:
A sample of five professors is taken to find how many Statistics books they have written. Find the sample standard deviation, if we know that they wrote respecitvely: 2, 3, 2, 5, 6 books. Round your answer to two decimals.

Practice Problem 2:
Last year, profits of five banks were `\$150000`, `\$130000`, `\$140000`, `\$100000`, `\$180000`. By using the tabular method, find the standard deviation of the profit. Round your answer to two decimals.

Practice Problem 3:
In the data set below, what is the sample standard deviation?
1, 2, 6, 67, 8, 5, 9

The standard deviation calculator, formula, step by step calculation using simple method, real world problems and practice problems would be very useful for grade school students (K-12 education) to learn what is standard deviation of a data set in statistics and probability, how to find it. It’s applications in real world is of great significance.

What is the formula of standard deviation?
The formula for Standard Deviation
s = √[{(x1-x̄)²+(x2-x̄)²+(x3-x̄)²+ . . . +(xn-x̄)²)}/(n-1)]
s - Sample Standard Deviation
x̄ - Sample Mean
n - number of members of sample

What is the difference between sample and population standard deviation?
The difference is the amount of data inclusion. In population standard deviation, the experiment includes the complete (finite number) number of data for analysis. The best example is employees data of an organization, students data of school or college etc. In sample standard deviation, the experiment includes the sample set of data for analysis. For example, it applies to situation where collecting the complete set of data is not possible.

Why standard deviation is so important in statistics?
Standard deviation tells how the data spread is around the mean. The difference between the each value of a sample and the mean is called the deviation. In other words, it tells how the common characteristics of members in a group varies to each other. This statistical model is used in almost every fields such as climate forecast, finance market research, real estate market analysis, material science, pharmaceuticals, clinical experiments etc. Standard deviation do help researchers to analyze and forecast the events through experiments, even in situations where collecting the entire set of data is not practically possible.