Probability and Statistics > Basic Statistics > How to find a five-number summary in statistics

**Contents**:

## How to find a five-number summary in statistics: Overview

The five number summary includes 5 items:

- The minimum.
- Q1 (the first quartile, or the 25% mark).
- The median.
- Q3 (the third quartile, or the 75% mark).
- The maximum.

The five number summary gives you a rough idea about what your data set looks like. for example, you’ll have your lowest value (the minimum) and the highest value (the maximum). Although it’s useful in itself, the main reason you’ll want to find a five-number summary is to find more useful statistics, like the interquartile range, sometimes called the middle fifty.

This how to article will guide you through how to find a five-number summary. Watch the video or read the steps below:

## How to Find a Five-Number Summary: Steps

**Step 1:***Put your numbers in ascending order*(from smallest to largest). For this particular data set, the order is:

Example: 1,2,5,6,7,9,12,15,18,19,27.**Step 2:***Find the minimum and maximum*for your data set. Now that your numbers are in order, this should be easy to spot.

In the example in step 1, the minimum (the smallest number) is 1 and the maximum (the largest number) is 27.**Step 3:***Find the median*. The median is the middle number. If you aren’t sure how to find the median, see: How to find the mean mode and median.**Step 4:***Place parentheses around the numbers***above and below**the median.

(This is not*technically*necessary, but it makes Q1 and Q3 easier to find).

(1,2,5,6,7),9,(12,15,18,19,27).**Step 5:***Find Q1 and Q3*. Q1 can be thought of as a median in the lower half of the data, and Q3 can be thought of as a median for the upper half of data.

(1,2,**5**,6,7)**, 9**, ( 12,15,**18**,19,27).**Step 6:***Write down your summary found in the above steps*.

minimum=1, Q1 =5, median=9, Q3=18, and maximum=27.

*That’s it!
*

## When the Summary doesn’t exist

Sometimes, it’s impossible to find a five-number summary. In order for the five numbers to exist, your data set must meet these two requirements:

- Your data must be
**univariate**. In other words, the data must be a single variable. For example, this list of weights is one variable: 120, 100, 130, 145. If you have a list of ages and you want to compare the ages to weights, it becomes bivariate data (two variables). For example: age 1 (25 pounds), 5 (60 pounds), 15 (129 pounds). The matching pairs makes it impossible to find a five number summary. - Your data must be ordinal, interval, or ratio.

Like the explanation? Check out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step solutions, just like this one!

## Box and whisker chart

A box and whiskers chart is a visual representation of the summary.

## Box Plot / Find a Five-Number Summary on the TI 89

When you create a box and whiskers chart on the

**TI-89**, the TI-89 will automatically calculate the five number summary for you.

**Sample problem:** Create a box and whiskers chart and find the five number summary for the following data: 200, 350, 300, 350, and 400.

**Step 1:** Create a new folder called “Box.” From the HOME screen, press F4 and scroll down to **NewFold** (option B). Press ENTER.

**Step 2:** Press 2nd Alpha ( – x to spell B O X and press ENTER.

**Step 3:** Press APPS, then scroll down to **Stats/List Editor**. Press ENTER twice.

**Step 4:** Press the down arrow key to get to the first line of the list. Enter your data into list1. Follow each entry with a comma: 200, 350, 300, 350, 400.

**Step 5: **Press F2 then 1 to enter **Plot Setup**.

**Step 6:** Press F1, right arrow, and 5 to select **mod box plot**.

**Step 7:** Arrow down to **Mark** and select **box**.

**Step 8:** Arrow down and enter B O X (using the alphanumeric keypad) in the **x**. Press ENTER.

**Step 9:** Read the boxplot. Press F3 and use the left and right cursors to find Min(200), Q1(250), Med(325), Q3(400), and Max(500).

That’s it!

**Tip**: if you want to change the folder back to MAIN, press MODES, scroll down to **Current Folder**. Press right key, then press 1 ENTER.

**Tip**If you get the error message **undefined variable**, it can be a frustrating process to try and solve the problem. Clearing the memory *may* help, but an easier way to get the box plot to graph is to enter the data into “list 1” in the List Editor and then type “list 1” as your “x” when defining the box plot.

Lost your guidebook? You can download a new one from the TI website here.

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## Find a Five Number Summary in SPSS

Calculating the five number summary is pretty straight forward if you have a small data set, but for larger data sets — which you will typically work with in SPSS — the task can be impossibly tedious. That’s where software like SPSS comes in handy — tasks that would sometimes take hours by hand can be calculated in a **fraction of a second. **The SPSS five number summary is calculated with the “Frequencies” tool.

Watch the video or read the steps below:

Step 1: **Open a new data sheet and type your data **into a column (or several columns). To open a new data sheet, click “File” in the toolbar, then click “new” and then click “Data.” Make sure you type your data without spaces (in other words, don’t leave empty rows).

Step 2: **Click “Analyze,”** then click “Descriptive Statistics” and then click “Frequencies” to open the Frequencies dialog box.

Step 3: **Click a variable name** (or several if you have entered your data into multiple columns) and then click the central arrow to move them to the Variable(s) list box. Note that SPSS uses the term “Variables,” but all it really means is the column header name. You can change this name by clicking the “Variables” view button at the bottom of the sheet.

Step 4: **Click “Statistics” **to open the Statistics dialog box.

Step 5: Check “Quartiles,” “median,” “mimimum” and “maximum” and then click “Continue.”

Step 6: **Click “OK”.** The SPSS five number summary is calculated and the results are returned in a new window.

**Note**: SPSS lists the first quartile (Q1) as the 25th percentile in the results window, and the third quartile (Q3) is listed as the 75th percentile.

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Now these questions I didn’t mind doing and they weren’t so mind bogging. I actually understood something for once. These kinds of problems anyone can do them, the information above is a lot more helpful then our book.

Even though this problem is kind of long, the problem and explanation itself was easy to do and easy to understand.

this helped me a million

i dont know why they dont teach it like this all the time,

simple

Thanx this is so much easier to remember I have a test on all this tommorrow so this is excellent help ;)

Thanks!!! :)

What about outliers?

In the *real* world, you’d probably discard them, if they look like statistical anomalies. On a test in class though, I’d go ahead and include them ;)

Stephanie

Really helpful to me, 50,000 times

This short explaination really helped.Thanks

So much easier than having to go book to my school books to have a look! Thanks.

thnxs but im still confused

Thanks so much for the easy explanation! I have been trying to get this and my teacher is of no help.

what happens when you break the numbers down to Q3 and Q1 but there are 2 medians?

Matt,

If you could post a specific example on the forum (click the tab up there ^^) then we’ll be better able to answer that.

Best,

Stephanie

Hi,

what is the minimum number for n(number of elements) to find the quartile.

I mean even if there are two elements, can there be quartile for that data

You *could* calculate quartiles. For example, median(q2) for for 1,2 = 1.5. If you want to calculate q1 and q3 though, you really need a minimum of four data points otherwise it becomes nonsensical.

One of the best ways to find the 5-number summary is by using the boxplot

I learned something today