**Contents: Standard Deviation **(click to skip to section):

**Basics:**

**More advanced topics:**

- Standard Deviation for a
**Binomial** **Discrete Random Variable Standard Deviation**- Standard Deviation for a
**Frequency Distribution**

**Using Technology:**

**Related articles:**

## Definition

Standard deviation is a measure of dispersement in statistics. “Dispersement” tells you how much your data is spread out. Specifically, it shows you how much your data is spread out around the mean or average. For example, are all your scores close to the average? Or are lots of scores way above (or way below) the average score?

### What Does it Look Like on a Graph?

The bell curve (what statisticians call a “normal distribution“) is commonly seen in statistics as a tool to understand standard deviation.

The following graph of a normal distribution represents a great deal of data in real life. The mean, or average, is represented by the Greek letter μ, in the center. Each segment (colored in dark blue to light blue) represents one standard deviation away from the mean. For example, 2σ means two standard deviations from the mean.

## Real Life Example

A normal distribution curve can represent hundreds of situations in real life. Have you ever noticed in class that most students get Cs while a few get As or Fs? That can be modeled with a bell curve. People’s weights, heights, nutrition habits and exercise regimens can also be modeled with graphs similar to this one. That knowledge enables companies, schools and governments to make predictions about future behavior. For behaviors that fit this type of bell curve (like performance on the SAT), you’ll be able to predict that 34.1 + 34.1 = 68.2% of students will score very close to the average score, or *one standard deviation away from the mean.*

## How to Find the Sample Standard Deviation by Hand

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## Standard Deviation for a Binomial

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Standard Deviation For a Binomial: TI-83

Standard Deviation For a Binomial: by hand

### TI 83 Standard Deviation For a Binomial

The TI 83 doesn’t have a built in function to find the standard deviation for a binomial. You have to enter the equation in manually.

**Example problem**: Find standard deviation for a binomial distribution with n = 5 and p = 0.12.

**Step 1:** Subtract p from 1 to find q.

1 – .12 ENTER

**=.88**

**Step 2:** Multiply n times p times q.

5 * .12 * .88 ENTER

**=.528**

**Step 3:** Find the square root of the answer from Step 2.

√.528 = **=.727** (rounded to 3 decimal places).

## Standard Deviation For a Binomial: By Hand

A binomial distribution is one of the simplest types of distributions in statistics. It’s a type of distribution where there is either success, or failure. For example, winning the lottery: or not winning the lottery. You can find the standard deviation for a binomial distribution in two ways:

- With a formula
- With a probability distribution table (scroll down for the steps)

The formula to find the standard deviation for a binomial distribution is:

Watch the video or read the steps below:

### Example question:

Find the standard deviation for the following binomial distribution: flip a coin 1000 times to see how many heads you get.

Step 1: Identify n and p from the question. N is the number of trials (given as 1000) and p is the probability, which is .5 (you have a 50% chance of getting a heads in any coin flip).

*At this point you can insert those numbers into the formula and solve. If formulas aren’t your forte, follow these additional steps:*

Step 2: Multiply n by p:

1000 * .5 = 500.

Step 3: Subtract “p” from 1:

1 – .5 = .5.

Step 4: Multiply Step 2 by Step 3: 500 * .5 = 250.

Step 5: Take the square root of Step 4:

√ 250 = 15.81.

*That’s it!*

## Standard Deviation of Discrete Random Variables

With discrete random variables, sometimes you’re given a probability distribution table instead of “p” and “n”. As long as you have a table you can calculate the standard deviation of discrete random variables with this formula:

**Example question: **Find the standard deviation of the discrete random variables shown in the following table, which represents flipping three coins:

Step 1: Find the mean (this is also called the expected value) by multiplying the probabilities by x in each column and adding them all up:

μ = (0 * 0.125) + (1 * 0.375) + (2 * 0.375) + (3 * 0.125) = 1.5

Step 2: work the inner part of the above equation, without the square root:

- ((0 – 1.5)
^{2}* 0.125 ) + - ((1 – 1.5)
^{2}* 0.375 ) + - ((2 – 1.5)
^{2}* 0.375 ) + - ((3 – 1.5)
^{2}* 0.125 ) + - = 0.75

Step 3: Take the square root of Step 2:

σ = √ 0.75 = 0.8660254.

*That’s it! *

## Standard Deviation for a **Frequency Distribution**

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The formula to find the standard deviation for a frequency distribution is:

Where:

- μ is the mean for the frequency distribution,
- f is the individual frequency counts,
- x is the value associated with the frequencies.

If formulas aren’t your forte, watch this short video, which shows you how to work the formula:

## How to find the Standard Deviation in Minitab

Watch the video or follow the steps below:

**Example question:** Find the standard deviation in Minitab for the following data: 102, 104, 105, 110, 112, 116, 124, 124, 125, 240, 245, 254, 258, 259, 265, 265, 278, 289, 298, 311, 321, 321, 324, 354

Step 1: **Type your data into a single column** in a Minitab worksheet.

Step 2: **Click “Stat”, then click “Basic Statistics,” then click “Descriptive Statistics.”**

Step 3: **Select the variables you want to find the standard deviation for **and then click “Select” to move the variable names to the right window.

Step 4: **Click the “Statistics” button.**

Step 5: **Check the “Standard deviation” box **and then click “OK” twice. The standard deviation will be displayed in a new window.

*That’s it!*

## How to find the Standard Deviation in SPSS

The tool to calculate **standard deviation in SPSS** is found in the “Analytics > Descriptive Statistics” section of the toolbar. You can also use the “Frequencies” option in the same menu. The video below shows both options, or read below for the steps with the first option only.

If you have already typed in your data into a worksheet, skip to Step 3.

Step 1: **Open a new worksheet to type in data.** Once SPSS opens, select the “type in data” radio button to the right of the “What would you like to do” dialog box.

Step 2: **Type your data into the worksheet.**You can use as many columns as you like to enter data, but don’t leave any blank rows between your data.

Step 3: **Click “Analyze” on the toolbar and then mouse over “Descriptive Statistics.” Click “Descriptives” to open the variables dialog box**.

Step 4: **Select the variables you want to find descriptive statistics for**. SPSS needs to know where the data is that you want to calculate the standard deviation for. The system will populate the left box with possibilities (columns of data that you entered) but you will need to select which variables you want to include and transfer those lists to the right box. To transfer the lists, click the center arrow to move those variables from the left box to the right box.

Step 5: Check the “Standard Deviation” box, then click “OK”. The answer will show to the right of the window, in the last column headed “std deviation.”

## References

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.

Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.

Lindstrom, D. (2010). Schaum’s Easy Outline of Statistics, Second Edition (Schaum’s Easy Outlines) 2nd Edition. McGraw-Hill Education

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