Probability and Statistics > Basic Statistics > Standard Deviation

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

- Standard Deviation
**Definition** - Standard Deviation
**Symbol** - How to
**Find**the Sample Standard Deviation by Hand - Standard Deviation for a
**Binomial** - Standard Deviation of
**Discrete Random Variables** - Standard Deviation for a
**Frequency Distribution**

**Using Technology:**

- How to find the Standard Deviation in
**Minitab** - How to find the Standard Deviation in
**SPSS** - Standard Deviation on the
**TI-89**Calculator - Standard Deviation on the
**TI-83**Calculator

## Definition

Standard deviation is a measure of dispersement in statistics. “Dispersement” just means 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.*

### What Does it Actually Mean?

It tells you how tightly your data is clustered around the mean. When the bell curve is flattened (your data is spread out), you have a large standard deviation — your data is further away from the mean. When the bell curve is very steep, your data has a small standard deviation — your data is tightly clustered around the mean. For example, the graph on the left might represent an abnormally high number of students getting scores close to the average, while the graph on the right represents more students getting scores *away* from the average.

## What is the Standard Deviation Symbol?

Which symbol you use depends on if you have a sample or a population:

- The symbol for a sample is
**s**. - The symbol for a population is σ

## How to Find the Sample Standard Deviation by Hand

Watch the video, or read the steps below for an alternate way:

The Σ means “to add up”, so what you’re basically doing to find the sample standard deviation is adding your numbers, squaring them and dividing. You can *easily* make errors calculating the sample standard deviation by hand. Check your work with our online variance and standard deviation calculator (it will give you all of the working out!) or use a TI-83 calculator to find the standard deviation (calculators *are* allowed for the AP statistics exam and most college professors also allow you to use one for tests. This is also one of the AP Statistics formulas) you’ll find on your formula sheet.

**Sample Problem: **Find the sample standard deviation for the following set of numbers: 12, 15, 17, 20, 30, 31, 43, 44, 54.

Step 1: Add up the numbers in the data set: 12 + 15 + 17 + 20 + 30 + 31 + 43 + 44 + 54 = 266.

Step 2: Square your answer and then divide by the number of items in your data set. There are 9 items in our example so:

266 x 266 = 70756 (squaring)

70756 / 9 = 7861.777777777777 (dividing by n)

Set this number aside for a moment.

Step 3: Take your set of original numbers from Step 1, and square them individually this time before you add them up:

(12 x 12) + (15 x 15) + (17 x 17) + (20 x 20) + (30 x 30) + (31 x 31) + (43 x 43) + (44 x 44) + (54 x 54) = 9620

Step 4: Subtract the amount in Step 2 from the amount in Step 3.

9620 – 7861.777777777777 = 1758.2222222222226

Set this number aside for a moment.

Step 5: Subtract 1 from the number of items in your data set. For our example:

9 – 1 = 8

Step 6: Divide the number in Step 4 by the number in Step 5. This gives you the variance:

1758.2222222222226 / 8 = 219.77777777777783

Step 7: Take the square root of your answer from Step 6. This gives you the standard deviation:

√(219.77777777777783) = 14.824903971958058

That’s it!

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## Standard Deviation for a Binomial

(Click to Skip to Section)

Standard Deviation For a Binomial: TI-83

Standard Deviation For a Binomial: by hand

### Standard Deviation For a Binomial: TI 83

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.

**Note**: If you’re finding the standard deviation for a binomial, you may also want to find the mean. I’ve included instructions for finding the mean for a binomial distribution at the bottom of this section; it only takes a single step more.

**Sample 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).

### Mean for a Binomial Distribution on the TI-83

**Sample problem**: Find the mean for a binomial distribution with n = 5 and p = 0.12.

Again, the TI 83 doesn’t have a function for this. But if you know the formula (n*p), it’s pretty easy to enter it on the home screen.

**Step 1:** Multiply n by p.

5 * .12 ENTER

**=.6**

Hey, that was easy!

**Something to think about:** You may be wondering *why* it was so easy to calculate the mean. After all being asked to “calculate the mean for a binomial distribution” *sounds* scary. If you think about what a mean (or average) is, then you’ll see why it was so easy. In the sample question, n = 5 and p = 0.12. What is “n”? That’s the number of items. So imagine a list of 5 items with a certain score:

1 = 0.12

2 = 0.12

3 = 0.12

4 = 0.12

5 = 0.12

If you were asked to find the average score for those five items, you wouldn’t even have to do the math: it’s just 0.12, right? Finding the mean for a binomial distribution is just a little different: you add up all of the probabilities (0.12 + 0.12 + 0.12 + 0.12 + 0.12). Or a faster way, just multiply n by p.

## 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 binomial distribution is:

Watch the video or read the steps below:

**Sample 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:

**Sample 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: **Click 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.”

## Standard Deviation on the TI-89 Calculator

These instructions are for the TI-89 Calculator. Instructions for other models (i.e. the Titanium) may differ slightly. Watch the video, or read the steps below.

**Example problem:** Find the Standard Deviation for the following data set: 1,34,56,89,287,598,1001.

**Step 1: ** *Press the HOME key.*

The home key is on the left hand side, third button from the top.

**Step 2: ***Press the CATALOG key*.

It’s located below the APPS key in the top middle of your keypad.

**Step 3: **Scroll down to stdDev( using the down arrow key. Press ENTER.

**Step 4: ***Press 2nd, then an open parentheses “(“.*

The following should be on screen (note the curly bracket you just added): stdDev({

**Step 5: ***Enter your set of numbers*. Follow each number with a comma except for the last.

You should have the following displayed on screen: stdDev({1,34,56,89,287,598,1001

**Step 6: ***Press 2nd, then a closing parentheses “)” twice.*

This adds a parentheses and curly parentheses to close the expression so you have stdDev({1,34,56,89,287,598,1001}). Important! Your expression must look exactly like this, with both sets of parentheses and curly parentheses.

**Step 7: ***Press ENTER.* The answer displayed is 375.149.

**Tip**: If you have a radical or fractional expression on screen (2√1724030/7) and require a decimal, just press the diamond key(♦) and ENTER to get a decimal.

## Standard Deviation on the **TI-83** Calculator

**Sample problem**: Find the standard deviation for the heights of the top 12 buildings in London, England. The heights, (in feet) are: 800, 720, 655, 655, 625, 600, 590, 529, 513, 502, 502, 502.

**Step 1:** Enter the above data into a list. Press the STAT button and then press ENTER. Enter the first number (800), and then press ENTER. Continue entering numbers, pressing ENTER after each entry.

** Step 2:** Press STAT.

** Step 3:** Press the right arrow button (the arrow keys are located at the top right of the keypad) to select

**Calc**.

** Step 4:** Press ENTER to highlight

**1-Var Stats**.

** Step 5:** Press ENTER again to bring up a list of stats. The standard deviation (Sx) for the above list of data is

**96.57**feet (rounded to 2 decimal places).

*That’s how to find the standard deviation on the TI-83!*

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you’re are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.