# Sample Standard Deviation (σ_{n - 1}) Calculator, Formula, Example & Step by Step Calculation

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**Standard Deviation (σ _{n-1}) calculator** - step by step calculation, formulas, solved example problem to estimate average distance (

**σ**) of individual observations from the group mean online by using

**(n - 1)**formula method, supports large numbers of grouped or ungrouped sample data, direct input, Excel, CSV (.csv) file & text (.txt) file format input.

**Contents**

- Features
- Standard Deviation
- Sample size estimation
- Formulas
- Step by step calculation
- Solved example problem
- Notable factors
- Direct user input method
- CSV file upload method
- TEXT file upload method
- Excel file based calculation

Most of the traditional standard deviation calculators supports direct input and allow users to find the end results only but this calculator supports various input methods and allow users not only to estimate the value of mean, standard deviation and variance, but it also reveals the complete **step by step calculation** like how users generally do on the paper to estimate how widely an invidual sample is dispersed from the mean value for the given sample data of a population. The complete step by step mean, standard deviation & variance calculation along with the corresponding results may very useful for users to understand how the sample data values are being used in such calculations. Except CSV & text file format input method, users can get the complete step by step calculation for each and every calculation done through direct input method.

**Features:**

1. It provides results along with complete step by step calculation.

2. It supports large number of data set through bulk file upload.

3. It accepts CSV file format input.

4. It accepts TEXT file format input.

5. It accepts direct user inputs like copy and paste or typing the input values directly into the input box.

6. It never stores your data used for calculation.

**Standard Deviation:**

In statistics, *standard deviation* often abbreviated to **SD, is a numerical value used to indicate how widely individual values in a group vary**. It is a measure of the average distance of individual observations from the group mean often denoted by the Greek symbol **σ** or by the alphabet **s** or **SD**. In many experiments, (n - 1) based formula method is often being used instead of entire population data **n** as because getting the entire population data for every experiments is practically not possible at all. Therefore, it's popularly known as sample standard deviation due to its usage of partial data samples of a population data. For entire population data, use this *population standard deviation calculator using n method* to measure the average distance of individual observations from the group mean.

SD is a **Bell Shaped Curve** in a graph of normal distribution illustrates how much deviation or spread (how widely individual values in a group vary from the group mean) is occurred or can be occurred from the expected value or mean of data set. The mean is the central value or expected value of the data set or distribution as same as the average or arithmetic mean of a data set. The main purpose of estimating the sample standard deviation is to measure how widely the individual sample values are being dispersed from the mean. It's popularly being used in various applications such as stock market in finance, weather forecasting, digital & analog signal processing, telecommunication, product pricing, polling etc. The SD formula using the denominator as (n - 1) instead of n is preferred whenever the data samples are chosen randomly from the population data. This method of estimation is also known as corrected sample standard deviation or Bessel's correction. It's a widely used method in various experiments as because it is practically not possible to collect entire population data for every experiments.

For normally distributed data, the 68%, 95% and 99.7% of random samples of a population lies between -1σ to 1σ, -2σ to 2σ and -3σ to 3σ respectively and it can easily be remembered by *68-95-99.7 rule*. The numbers in the rule 68, 95 and 99.7 represents the percentage range or confidence interval of random samples of the population data for 1σ, 2σ and 3σ respectively. The precise percentage values of the band of 1σ, 2σ, and 3σ are 68.27%, 95.45% and 99.73% respectively. The lower SD values closer to expected value gives the precise estimation.

**Sample Size Estimation**

To design the experiment, the sample size estimation is much important to have the experiment results more generalised & have enough power. Therefore, users may use this *experiment sample size calculator* to determine how much sample data should be taken from the population data. This calculator may be very useful to determine how much samples required for designing the experiments like clinical data or strength of material data or similar kind of data analysis.

__Formulas__

The below mathematical formulas are used to find mean, standard deviation & variance for any sample data of a population. Refer the solved example for step by step calculation.

__Step by step calculation__

The below step by step calculation help users to understand how to perform standard deviation.

1. find the number of samples to be taken from a data set.

2. find the sum of all the individual sample values by using the formula **∑ ^{n}_{i=1} x_{i} = x_{1} + x_{2} + x_{3} + ... + x_{n}**

3. find the mean by using the formula

**x̄ = ∑**

^{n}_{i=1}x_{i}/n4. find the standard deviation by using the formula

**σ**

_{n-1}= √[∑^{n}_{i=1}(x_{i}- x̄)^{2}/ (n - 1)]5. Find the variance by taking

**(σ**square of sample sd σ.

^{2})The manual estimation can be done by using the above formulas & steps to find the sample deviation for any distribution. However, to make the estimate simple as possible, this featured online calculator may useful to estimate sample standard deviation for large numbers of grouped or ungrouped samples. The estimate can be done in three ways by using this calculator as it supports direct user input, csv file upload and text file format input.

__Solved Example Problem__

Solved example to estimate the sample deviation from the mean shows how the population data values are being used in the calculation based on the sample standard deviation formulas.

**Example Problem :**

Calculate the mean, SD and variance for random samples of a population 7, 9, 8, 6, 7, 12 and 10

**Solution:**

Mean = (7 + 9 + 8 + 6 + 7 + 12 + 10)/7

= 59/7

= 8.42857

σ_{(n - 1)} = √( (1/7-1) * (7-8.42857)^{2} + (9-8.42857)^{2} + (8-8.42857)^{2} + (6-8.42857)^{2} + (7-8.42857)^{2} + (12-8.42857)^{2} + (10-8.42857)^{2})

= √( (1/6) * (-1.42857^{2} + 0.57143^{2} + -0.42857^{2} + -2.42857^{2} + -1.42857^{2} + 3.57143^{2} + 1.57143^{2}))

= √( (1/6) * (2.0408122449 + 0.3265322449 + 0.1836722449 + 5.8979522449 + 2.0408122449 + 12.7551122449 + 2.4693922449))

= √4.2856866361

**σ _{(n - 1)}** = 2.07019

variance = σ

^{2}

= 2.07019 x 2.07019

= 4.28568

**Notable Factors :**

1. Estimating σ is used to analyze the statistical data.

2. No standard deviation exists (σ = 0) if each data of a data set contains the same number. Also σ is not applicable for less than 2 real numbers as the value of σ = 0.

3. The σ can be calculated for real numbers only.

4. Low σ value closer to expected value (mean) gives more precise estimation

5. Higher σ value which deviates more from the expected value (mean) gives relatively lesser precise estimation

6. In normal distribution, the σ is always a Bell Shaped Curve as the deviation or dispersion equally falls both sides of the sample mean or expected value

7. **68.27%** values lie within -1σ to 1σ of the mean in normal distribution.

8. **95.45%** values lie within -2σ to 2σ of the mean in normal distribution.

9. **99.73%** values lie within -3σ to 3σ of the mean in normal distribution.

__Direct User Input Method:__

The main purpose of this direct user input method is to find the sample standard deviation for small random set of data and to generate the complete step by step average uncertainty calculation. In this calculator, users may directly supply the input values. For larger set of data, either CSV or TXT file format input method is recommended.

**For ungrouped data**

1. Copy and paste your data set in the input box. Validate and make sure the data copied only contains real numbers and comma separated.

2. Hit on the calculate button provides sample mean, SD, variance and the complete step by step calculation.

3. Note down the results and the complete step by step calculation.

**For grouped data**

1. Copy and paste the first group data in the input box. Validate and ensure that the data copied only contains real numbers and comma separated.

2. Hit on the calculate button provides mean, SD, variance and the complete calculation for the first group.

3. Note down the results for the first group.

4. If more groups exist, copy paste the second group data in the input box.

5. Hit on the calculate button provides sample mean, SD, variance and the complete calculation for the second group.

6. Note down the results.

7. If more groups exist, repeat the same steps until the last group of data.

__CSV file Upload Method:__

**For Ungrouped Data**

The main purpose is to calculate the mean, standard deviation and variance for entire sample data once, irrespective to grouped or ungrouped data in the CSV file. The maximum CSV file size is 2 MB. This calculator for ungrouped data requires only one column data for input. Therefore, the CSV file must have first column as data set. Any row containing the empty data in the CSV file will be ignored.

1. Validate your CSV file (.csv file extension) data and ensure that the file not contains non-real numbers except first row and the file size is not exceeding more than 2 MB. The first row may contain either column name or data. If it contains the column name, it should be ignored by selecting the **Skip first row** option.

2. The first column data is the key column of the CSV file, must be the data set and real numbers. Only one column data values will be taken into account to calculate the sample deviation value of the data. By default, this calculator assumes your first column data as your input data. Therefore user should customize CSV file to have the first column as the primary data of the calculation. All remaining columns of your csv file will be ignored in the calculation, if any.

3. Select **Ungrouped Data** to calculate SD for entire data once.

4. Select **Yes** to ignore first row otherwise select **No**.

5. Choose the preferred CSV file containing the data.

6. Hit on upload to calculate the SD.

7. Note down the Results.

**For Grouped Data**

The main purpose is to group the data based on its nature and calculate the sample mean, standard deviation and variance for each group separately. Any grouped or ungrouped data of the CSV file that is uploaded will be sorted and grouped each time by this calculator itself. Therefore, the separate grouping of data is not necessary to calculate the standard deviation for group data. The maximum CSV file size is 2 MB. This calculator for grouped data requires two columns data for input. Therefore, the CSV file must have first column as data set and real numbers and the second column must be the group data column. All remaining columns in the csv file will be ignored in the calculation by default. Any row containing the empty data in the CSV file will be ignored.

1. Validate your csv file data and ensure that the file not contains non-real numbers except first row and the file size is not exceeding more than 2 MB. The first row may contain either column name or data. If it contains the column name, it should be ignored by selecting the **Skip first row** option.

2. In the CSV file, the first column data must be the data set and real numbers and the second column data must be respective category data.

3. Select **Grouped Data** to calculate SD for entire data once.

4. Select **Yes** to ignore first row otherwise select **No**.

5. Choose the preferred CSV file containing the data.

6. Hit on upload to calculate the SD.

7. Note down the Results for each and every group.

__TEXT file Upload Method:__

**For Ungrouped Data**

The main purpose is to calculate the sample mean, standard deviation and variance for entire data only once irrespective to whether the data is grouped or ungrouped in the TEXT file. The maximum TXT file size is 2 MB. This calculator for ungrouped data requires only one column data for input. By default, this calculator assumes your first column data as your input data. Therefore user should customize CSV file to have the first column as the primary data of the calculation. All remaining columns in the file will be ignored in the calculation, if any. Any row containing the empty data in the TXT file will be ignored.

1. Validate your TEXT file (.txt file extension) data and ensure that the file not contains non-real numbers except first row and the file size is not exceeding more than 2 MB. The first row may contain either column name or data. If it contains the column name, it should be ignored by selecting the **Skip first row** option.

2. The first column data is the key part of the preferred TXT file, must be the data samples and real numbers.

3. Select **Ungrouped Data** to calculate SD for entire data only once.

4. Select **Yes** to ignore first row otherwise select **No**.

5. Carefully choose the data **Delimiter**.

6. Choose the preferred TXT file containing the data.

7. Hit on upload to calculate the SD.

8. Note down the Results.

**For Grouped Data**

The main purpose is to group the data based on its nature and calculate the sample mean, standard deviation and variance for each group separately. Any grouped or ungrouped data of the TEXT file that is uploaded will be sorted and grouped each time by this calculator itself. Therefore, the separate grouping of data is not necessary to calculate the standard deviation for group data. The maximum TXT file size is 2 MB. This calculator for grouped data requires two columns data for input. By default, this calculator assumes your first column containing data for SD and the second column data containing the group information. The grouping will be done based on the second column data. Therefore user should customize the TEXT file to have the first column as the population data and the second column as group information data. All remaining columns in the TXT file will be ignored in the calculation, if any. Any row containing the empty data in the TXT file will be ignored.

1. Validate your TEXT file data and ensure that the file not contains non-real numbers except first row and the file size is not exceeding more than 2 MB. The first row may contain either column name or data. If it contains the column name, it should be ignored by selecting the **Skip first row** option.

3. Select **Grouped Data** to calculate SD for entire data once.

4. Select **Yes** to ignore first row otherwise select **No**.

5. Carefully choose the data **Delimiter**.

6. Choose the preferred TXT file containing the data.

7. Hit on upload to calculate the SD.

8. Note down the Results for each and every group.

__Excel File Based Calculation:__

The Excel file data can't be directly supported by this calculator. The Excel file have to be converted into CSV file format first. To convert Excel to CSV and calculate the standard deviation, follow the below steps

**File Conversion based Calculation**

1. Open and Click on the Excel office button provided in the top most left side corner.

2. Click on Save As opens the new window to locate the directory, name the file and select the preferred file type to save.

3. Navigate to the directory where you want to save the file.

4. Type the preferred File Name.

5. Select the Save as type as CSV (comma delimited) (*.csv).

6. Follow the same instruction given for CSV file format input and standard deviation calculation to find the sample data dispersion of the experiment.

The usage of SD is high in demand and spread across various fields such as measuring risk of investments (stocks, bonds, mutual funds, etc.) in finance, forecasting the weather, digital & analog signal processing, product pricing, telecommunication, polling etc. In *probability & statistics*, while predicting the individual sample distance from group mean, calculating the sample standard deviation is very important. A larger dispersion value of a individual data spread occurs too far from the mean whereas the smaller **σ** value indicates that the individual data spread is occurred very close to the mean of the sample data set. When it comes to the measure how widely the individual values in a group vary from the average value, this online sample standard deviation calculator may useful to perform & verify such calculations to find group mean, individual data average distance from group mean & variance.