Quartiles are the values that divide a list of numbers into quarters: Show
Like this: Example: 5, 7, 4, 4, 6, 2, 8Put them in order: 2, 4, 4, 5, 6, 7, 8 Cut the list into quarters: And the result is:
Sometimes a "cut" is between two numbers ... the Quartile is the average of the two numbers. Example: 1, 3, 3, 4, 5, 6, 6, 7, 8, 8The numbers are already in order Cut the list into quarters: In this case Quartile 2 is half way between 5 and 6: Q2 = (5+6)/2 = 5.5 And the result is:
Interquartile RangeThe "Interquartile Range" is from Q1 to Q3: To calculate it just subtract Quartile 1 from Quartile 3, like this: Example:The Interquartile Range is: Q3 − Q1 = 7 − 4 = 3 Box and Whisker PlotWe can show all the important values in a "Box and Whisker Plot", like this: A final example covering everything: Example: Box and Whisker Plot and Interquartile Range for4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11 Put them in order: 3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18 Cut it into quarters: 3, 4, 4 | 4, 7, 10 | 11, 12, 14 | 16, 17, 18 In this case all the quartiles are between numbers:
Also:
So now we have enough data for the Box and Whisker Plot: And the Interquartile Range is: Q3 − Q1 = 15 − 4 = 11 The interquartile range is a measure of where the “middle fifty” is in a data set. Where a range is a measure of where the
beginning and end are in a set, an interquartile range is a measure of where the bulk of the values lie. That’s why it’s preferred over many other measures of spread when reporting things like school performance or SAT scores. The interquartile range formula is the first
quartile subtracted from the third quartile: Watch the video for how to calculate the interquartile range by hand: How to find an interquartile range Can’t see the video? Click here. Contents:Solving by hand:
Using Technology:
General info:
Solve the formula by hand.Steps:
What if I Have an Even Set of Numbers?Example question: Find the IQR for the following data set: 3, 5, 7, 8, 9, 11, 15, 16, 20, 21.
Back to Top Find an interquartile range for an odd set of numbers: Alternate MethodAs you may already know, nothing is “set in stone” in statistics: when some statisticians find an interquartile range for a set of odd numbers, they include the
median in both both quartiles. For example, in the following set of numbers: 1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27 some statisticians would break it into two halves, including the median (9) in both halves: Box Plot interquartile range: How to find itWatch the video for the steps.
How to find an interquartile range on a boxplot Can’t see the video? Click here. Box Plot interquartile range: How to find itExample question: Find the interquartile range for the above box plot.
That’s it! Back to Top Interquartile Range in Minitab
How to find an interquartile range in Minitab Can’t see the video? Click here. Interquartile Range in Minitab: StepsExample question: Find an interquartile range in Minitab for the Grade Point Average (GPA) in the following data set: Step 1: Type your data into a Minitab worksheet. Enter your data into one or two columns. Step 2: Click “Stat,” then click “Basic Statistics,” then click “Display Descriptive Statistics” to open the Descriptive Statistics menu. Step 3: Click a variable name in the left window and then click the “Select” button to transfer the variable name to the right-hand window. Step 4: Click the “Statistics” button. Step 5: Check “Interquartile Range.” Step 6: Click the “OK” button (a new window will open with the result). The IQR for the GPA in this particular data set is 1.8. That’s it! Tip: If you don’t see descriptive statistics show in a window, click “Window” on the toolbar, then click “Tile.” Click the Session window (this is where descriptive statistics appear) and then scroll up to see your results. Back to Top Interquartile Range in Excel 2007How to Find an Interquartile Range Excel 2007Watch the video or read the steps below to find an interquartile range in Excel 2007: How to find an interquartile range in Excel Steps: Step 2: Click a blank cell (for example, click cell B2) and then type =QUARTILE(A2:A10,1). You’ll need to replace A2:A10 with the actual values from your data set. For example, if you typed your data into B2 to B50, the equation is =QUARTILE(B2:B50,1). The “1” in this Excel formula(A2:A10,1) represents the first quartile (i.e the point lying at 25% of the data set). Step 3: Click a second blank cell (for example, click cell B3) and then type =QUARTILE(A2:A10,3). Replace A2:A10 with the actual values from your data set. The “3” in this Excel formula (A2:A10,3) represents the third quartile (i.e. the point lying at 75% of the data set). Step 4: Click a third blank cell (for example, click cell B4) and then type =B3-B2. If your quartile functions from Step 2 and 3 are in different locations, change the cell references. Step 5: Press the “Enter” key. Excel will return the IQR in the cell you clicked in Step 4 That’s it! Back to Top How to Find an Interquartile Range in SPSSLike most technology, SPSS has several ways that you can calculate the IQR. However, if you click on the most intuitive way you would expect to find it (“Descriptive Statistics > Frequencies”), the surprise is that it won’t list the IQR (although it will list the first, second and third quartiles). You could take this route and then subtract the third quartile from the first to get the IQR. However, the easiest way to find the interquartile range in SPSS by using the “Explore” command. If you have already typed data into your worksheet, skip to Step 3. Watch the video for the steps: How to find the Interquartile Range in SPSS Can’t see the steps? Click here. StepsStep 1: Open a new data file in SPSS. Click “File,” mouse over “New” and then click “Data.” Step 2: Type your data into columns in the worksheet. You can use as many columns as you need, but don’t leave blank rows or spaces between your data. See: How to Enter Data into SPSS. Step 3: Click “Analyze,” then mouse over “Descriptive Statistics.” Click “Explore” to open the “Explore” dialog box. Step 4: Click the variable name (that’s just a fancy name for the column heading), then click the top arrow
to move the variable into the “Dependent list” box. Step 5: Click “OK.” The interquartile range is listed in the Descriptives box. Tip: This example has only one list typed into the data sheet, but you may have several to choose from depending on how you entered your data. Make sure you select the right variable (column names) before proceeding. If you want more memorable variable names, change the column title by clicking the “variable view” button at the very bottom left of the worksheet. Type in your new variable name and then return to data view by clicking the “data
view” button. What is an Interquartile Range?
If you want to know that the IQR is in formal terms, the IQR is calculated as: The difference between the third or upper quartile and the first or lower quartile. Quartile is a term used to describe how to divide the set of data into four equal portions (think quarter). IQR ExampleIf you have a set containing the data points 1, 3, 5, 7, 8, 10, 11 and 13, the first quartile is 4, the second quartile is 7.5 and the third quartile is 10.5. Draw these points on a number line and you’ll see that those three numbers divide the number line in quarters from 1 to 13. As such, the IQR of that data set is 6.5, calculated as 10.5 minus 4. The first and third quartiles are also sometimes called the 25th and 75th percentiles because those are the equivalent figures when the data set is divided into percents rather than quarters. Back to Top Interquartile Range using the TI83Watch the video for the steps: TI 83 Interquartile Range Can’t see the video? Click here. While you can use the nifty online interquartile range calculator on this website, that might not be an option in a quiz or test. Most instructors allow the use of a TI-83 on tests, and it’s even one of the few calculators allowed in the AP Statistics exam. Finding the TI 83 interquartile range involves nothing more than entering your data list and pushing a couple of buttons. Example problem: Find the TI 83 interquartile range for the heights of the top 10 buildings in the world (as of 2009). The heights, (in feet) are: 2717, 2063, 2001, 1815, 1516, 1503, 1482, 1377, 1312, 1272. StepsStep 1: Enter the above data into a list on the TI 83 calculator. Press the STAT button and then press ENTER. Enter the first number (2717), and then press ENTER. Continue entering numbers, pressing ENTER after each entry. Step 2: Press the STAT button. 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. Step 6:Scroll down the list with the arrow keys to find Q1 and Q3. Write those numbers down. You could copy and paste the numbers but unfortunately, Texas Instruments doesn’t make this easy:
The copy and paste menu should appear, enabling you to copy and paste the data. You would have to do this twice (returning to the HOME screen each time), so it’s much faster just to write the numbers down. Step 7:Subtract Q1 from Q3 to find the IQR (strong>624 feet for this set of numbers). That’s it! How to Find Q1, Q3 and the Interquartile Range TI 89Example problem: Find Q1, Q3, and the IQR for the following list of numbers: 1, 9, 2, 3, 7, 8, 9, 2. Step 1: Press APPS. Scroll to Stats/List Editor (use the arrow keys on the keypad to scroll). Press ENTER. If you don’t have the stats/list editor you can download it here. Step 2: Clear the list editor of data: press F1 8. Step 3: Press ALPHA 9 ALPHA 1 ENTER. This names your list “IQ.” Step 4: Enter your numbers, one at a time. Follow each entry by pressing the ENTER key. For our group of numbers, enter Step 5: Press F4, then ENTER (for the 1-var stats screen). Step 6: Tell the calculator you want stats for the list called “IQ” by entering ALPHA 9 ALPHA 1 into the “List:” box. The calculator should automatically put the cursor there for you. Press ENTER twice. Step 7:Read the results. Q1 is listed as Q1X (in our example, Q1X=2). Q3 is listed as Q3X (Q3X=8.5). To find the IQR, subtract Q1 from Q3 on the Home screen. The IQR is 8.5-2=6.5. That’s it! Back to Top What is The Interquartile Range Formula?The IQR formula is: Where Q3 is the upper quartile and Q1 is the lower quartile. IQR as a test for normal distribution Use the interquartile range formula with the mean and
standard deviation to test whether or not a population has a normal distribution. The formula to determine whether or not a population is normally distributed are: Back to Top What is an Interquartile Range Used For?
Where Does the term Interquartile Range Come From?Who invented the term “Interquartile Range?” In order to find that out, we have to go back to the 19th century. HistoryBritish physician Sir Donald MacAlister used the terms lower quartile and higher quartile in the 1879 publication,
the Law of the Geometric Mean. Proc. R. Soc. XXIX, p. 374: ” “As these two measures, with the mean, divide the curve of facility into four equal parts, I propose to call them the ‘higher quartile’ and the ‘lower quartile’ respectively.” Although a physician by trade, he was gifted with mathematics and achieved the highest score in the final mathematics exams at Cambridge University in 1877. He spoke nineteen languages including English, Czech and Swedish. Macalister’s paper, the Law of the Geometric Mean was actually in response to a question put forward by Francis Galton (inventor of the Galton board). However, it wasn’t until 1882 that Galton (“Report of the Anthropometric Committee”) used the upper quartile and lower quartile values and the term “interquartile range” —defined as twice the probable error. Galton wasn’t just a statistician—he was also an anthropologist, geographer, proto-genetecist and psychometrician who produced more than 340 books. He also coined the statistical terms “correlation” and “regression toward the mean.” ReferencesGonick, L. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, pp. 20-21, 1993. ---------------------------------------------------------------------------
Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free! Comments? Need to post a correction? Please Contact Us. What is the formula to find Q1?When the set of observations are arranged in ascending order the quartiles are represented as, First Quartile(Q1) = ((n + 1)/4)th Term. Second Quartile(Q2) = ((n + 1)/2)th Term.
How do you find Q1 Q1 and Q3?There are four different formulas to find quartiles: Formula for Lower quartile (Q1) = N + 1 multiplied by (1) divided by (4) Formula for Middle quartile (Q2) = N + 1 multiplied by (2) divided by (4) Formula for Upper quartile (Q3) = N + 1 multiplied by (3) divided by (4)
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