What Is The Interquartile Range Of The Data

What is an Interquartile Range?

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 information technology's preferred over many other measures of spread when reporting things similar school performance or SAT scores.

The interquartile range formula is the first quartile subtracted from the third quartile:

IQR = Q₃ – Q_i.

Watch the video for how to summate the interquartile range by paw:

How to find an interquartile range

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Solve the formula by manus.

Steps:

Pace 1: Put the numbers in lodge.
1, two, 5, 6, 7, 9, 12, 15, xviii, 19, 27.
Step two: Notice the median.
1, two, 5, six, 7, 9, 12, 15, 18, nineteen, 27.
Step three: Place parentheses around the numbers higher up and below the median.
Non necessary statistically, but information technology makes Q1 and Q3 easier to spot.
(1, 2, 5, 6, seven), 9, (12, 15, 18, 19, 27).
Pace 4: Notice Q1 and Q3
Think of Q1 as a median in the lower half of the data and recollect of Q3 as a median for the upper half of data.
(i, 2, 5, 6, 7), 9, ( 12, xv, 18, xix, 27). Q1 = v and Q3 = 18.
Step five: Decrease Q1 from Q3 to find the interquartile range.
18 – v = 13.

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

What if I Have an Fifty-fifty Prepare of Numbers?

Example question: Find the IQR for the post-obit data prepare: 3, 5, 7, 8, 9, 11, 15, 16, xx, 21.

Footstep ane: Put the numbers in society.
3, 5, 7, 8, 9, xi, 15, sixteen, 20, 21.
Footstep 2: Make a mark in the center of the data:
3, 5, vii, eight, 9, | 11, 15, 16, 20, 21.
Step three: Place parentheses around the numbers above and beneath the mark you made in Step 2--it makes Q1 and Q3 easier to spot.
(3, 5, 7, eight, ix), | (11, 15, sixteen, xx, 21).
Step 4: Find Q1 and Q3
Q1 is the median (the eye) of the lower half of the data, and Q3 is the median (the middle) of the upper half of the data.
(3, 5, 7, 8, 9), | (11, 15, sixteen, 20, 21). Q1 = 7 and Q3 = sixteen.
Step v: Subtract Q1 from Q3.
16 – vii = ix.
This is your IQR.

Find an interquartile range for an odd prepare of numbers: Alternate Method

As you may already know, nothing is "fix in rock" in statistics: when some statisticians find an interquartile range for a ready of odd numbers, they include the median in both both quartiles. For example, in the following set of numbers: 1, 2, 5, half dozen, 7, 9, 12, 15, 18, 19, 27 some statisticians would break it into ii halves, including the median (9) in both halves:
(i, ii, 5, 6, 7, ix), (nine, 12, 15, 18, 19, 27)
This leads to two halves with an even set of numbers, and so you lot can follow the steps above to find the IQR.
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Box Plot interquartile range: How to find it

Watch the video for the steps.

How to find an interquartile range on a boxplot

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Box Plot interquartile range: How to find it

box plot interquartile range

Example question: Find the interquartile range for the above box plot.

Step i: Find Q1.Q1 is represented by the left hand edge of the "box" (at the point where the whisker stops).

In the above graph, Q1 is approximately at 2.6. (A complete explanation of Q1 is here: The five number summary.)
Step 2: Find Q3.
Q3 is represented on a boxplot past the correct hand edge of the "box".

Q3 is approximately 12 in this graph.
Step 3: Subtract the number yous constitute in pace 1 from the number you found in step 3.
This volition give you lot the interquartile range. 12 – two.half-dozen = 9.4.

That's information technology!

Interquartile Range in Minitab

Lookout the video for footstep-by-step directions:

How to detect an interquartile range in Minitab

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Interquartile Range in Minitab: Steps

Example question: Find an interquartile range in Minitab for the Grade Point Average (GPA) in the post-obit information set:
Grade Point Average (GPA): ane(3.2), one(three.1), 2(3.5), 2(2.0), 3(1.9), 3(iv.0), iii(3.ix), iv(3.8), 4(two.9), five(3.9), 5(three.2), five(3.iii), 6(3.iv), 6(2.half dozen), vi(2.v), seven(two.0), 7(1.5), 8(4.0), eight(ii.0).

Pace 1: Type your data into a Minitab worksheet. Enter your data into one or two columns.
minitab interquartile range a

Step 2: Click "Stat," then click "Basic Statistics," then click "Brandish Descriptive Statistics" to open the Descriptive Statistics carte du jour.
minitab interquartile range b

Footstep 3: Click a variable name in the left window and then click the "Select" button to transfer the variable name to the right-paw window.

Step iv: Click the "Statistics" button.
minitab interquartile range c

Pace 5: Cheque "Interquartile Range."
minitab interquartile range d .

Step vi: Click the "OK" button (a new window will open up with the consequence). The IQR for the GPA in this particular data set is 1.viii.
minitab interquartile range e

That'southward it!

Tip: If y'all don't see descriptive statistics testify in a window, click "Window" on the toolbar, so click "Tile." Click the Session window (this is where descriptive statistics appear) and then whorl upwardly to see your results.

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Interquartile Range in Excel 2007

How to Find an Interquartile Range Excel 2007

Watch the video or read the steps below to notice an interquartile range in Excel 2007:

How to find an interquartile range in Excel

Steps:
Step 1: Enter your data into a single Excel column on a worksheet. For instance, type your data in cells A2 to A10. Don't leave whatever gaps in your information.

Step 2: Click a bare cell (for example, click prison cell B2) and then type =QUARTILE(A2:A10,i). Yous'll need to supplant A2:A10 with the actual values from your data fix. For example, if y'all typed your information into B2 to B50, the equation is =QUARTILE(B2:B50,1). The "1" in this Excel formula(A2:A10,ane) represents the first quartile (i.e the bespeak lying at 25% of the data set).

Stride three: Click a 2d blank cell (for example, click jail cell B3) and so blazon =QUARTILE(A2:A10,3). Replace A2:A10 with the bodily values from your data fix. The "three" in this Excel formula (A2:A10,three) represents the third quartile (i.e. the point lying at 75% of the information set).

Step iv: Click a tertiary bare cell (for example, click cell B4) and then type =B3-B2. If your quartile functions from Step 2 and 3 are in different locations, alter the cell references.

Pace 5: Press the "Enter" central. Excel will return the IQR in the cell yous clicked in Pace 4

That's it!

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How to Notice an Interquartile Range in SPSS

Like most technology, SPSS has several means that you can summate the IQR. However, if you click on the well-nigh intuitive manner you would await to find it ("Descriptive Statistics > Frequencies"), the surprise is that it won't listing the IQR (although it volition list the first, 2nd and third quartiles). You could take this route and then subtract the tertiary quartile from the outset to get the IQR. Notwithstanding, the easiest way to find the interquartile range in SPSS past using the "Explore" command. If yous have already typed data into your worksheet, skip to Pace iii.

Spotter the video for the steps:

How to find the Interquartile Range in SPSS

Tin can't see the steps? Click hither.

Steps

Step 1: Open a new data file in SPSS. Click "File," mouse over "New" and so click "Data."

Step 2: Blazon your data into columns in the worksheet. You can use as many columns as you need, merely don't leave blank rows or spaces betwixt your data. See: How to Enter Data into SPSS.

Step iii: Click "Analyze," and then mouse over "Descriptive Statistics." Click "Explore" to open up the "Explore" dialog box.
how to find the interquartile range in SPSS 1

Pace 4: Click the variable name (that'south just a fancy name for the column heading), then click the top arrow to move the variable into the "Dependent list" box.

The "Explore" variables dialog box.

Stride v: Click "OK." The interquartile range is listed in the Descriptives box.
interquartile range SPSS 3

Tip: This example has just one list typed into the data sheet, just you may have several to choose from depending on how you entered your data. Make sure you select the right variable (cavalcade names) before proceeding. If you want more memorable variable names, change the column title by clicking the "variable view" push button at the very bottom left of the worksheet. Type in your new variable name and then render to information view by clicking the "data view" button.
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What is an Interquartile Range?

Imagine all the data in a set as points on a number line. For example, if you have 3, 7 and 28 in your set of data, imagine them equally points on a number line that is centered on 0 only stretches both infinitely beneath nix and infinitely higher up zero. Once plotted on that number line, the smallest data betoken and the biggest data point in the set of data create the boundaries (i.eastward. a lower jump and an upper leap) of an interval of space on the number line that contains all information points in the set. The interquartile range (IQR) is the length of the heart 50% of that interval of space.

what is an interquartile range

The interquartile range is the middle 50% of a data prepare. Box and whiskers image by Jhguch at en.wikipedia

If you want to know that the IQR is in formal terms, the IQR is calculated as: The difference between the 3rd or upper quartile and the beginning or lower quartile. Quartile is a term used to describe how to divide the set of data into 4 equal portions (remember quarter).

IQR Example

If you have a gear up containing the data points 1, three, five, 7, 8, x, xi and 13, the offset quartile is 4, the second quartile is 7.5 and the third quartile is x.five. Depict these points on a number line and yous'll meet 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.

Interquartile Range using the TI83

Scout the video for the steps:

TI 83 Interquartile Range

Tin't come across the video? Click here.

While you tin can use the nifty online interquartile range calculator on this website, that might non exist an option in a quiz or test. Most instructors allow the use of a TI-83 on tests, and information technology's even i of the few calculators allowed in the AP Statistics exam. Finding the TI 83 interquartile range involves nothing more than than entering your data list and pushing a couple of buttons.

Example problem: Notice 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.

Steps

Step ane: Enter the higher up data into a list on the TI 83 calculator. Press the STAT button and then press ENTER. Enter the offset number (2717), and then press ENTER. Continue entering numbers, pressing ENTER after each entry.

Step 2: Printing the STAT button.

Footstep 3: Press the right arrow button (the arrow keys are located at the top correct of the keypad) to select "Calc."

Step four: Press ENTER to highlight "1-Var Stats."

Step 5: Press ENTER again to bring upwards a listing of stats.

Step six: Scroll down the list with the arrow keys to notice Q1 and Q3. Write those numbers downwards. Y'all could copy and paste the numbers but unfortunately, Texas Instruments doesn't make this easy:

Use the arrow keys to place the cursor at the starting time of the
text that you desire to highlight.
Using the TI Keyboard, press and hold down the Shift key, and then use the arrow keys to highlight the text.
Release the Shift fundamental and arrow central.

The re-create and paste menu should appear, enabling you to copy and paste the data. You lot would have to exercise this twice (returning to the HOME screen each time), and so information technology'southward much faster only to write the numbers down.

Step 7:Decrease Q1 from Q3 to detect the IQR (potent>624 feet for this set of numbers).

That's information technology!
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How to Discover Q1, Q3 and the Interquartile Range TI 89

Case problem: Find Q_ane, Q₃, and the IQR for the following list of numbers: one, nine, 2, 3, 7, eight, 9, 2.

Step 1: Press APPS. Ringlet to Stats/Listing Editor (use the pointer keys on the keypad to ringlet). Printing ENTER. If you lot don't have the stats/listing editor y'all can download information technology here.

Step 2: Clear the listing editor of data: press F1 8.

Step 3: Press ALPHA nine ALPHA 1 ENTER. This names your list "IQ."

Footstep iv: Enter your numbers, one at a time. Follow each entry past pressing the ENTER key. For our grouping of numbers, enter
i,9,2,3,7,8,9,2

Step 5: Press F4, then ENTER (for the 1-var stats screen).

Pace 6: Tell the calculator you want stats for the list called "IQ" by inbound Alpha 9 Blastoff 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). Q_three is listed as Q3X (Q3X=8.5). To find the IQR, subtract Q_i from Q₃ on the Home screen. The IQR is 8.five-2=6.5.

That's it!

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What is The Interquartile Range Formula?

The IQR formula is:
IQR = Q₃ – Q₁

Where Q₃ is the upper quartile and Q_i is the lower quartile.

the interquartile range formula

IQR as a exam for normal distribution

Use the interquartile range formula with the mean and standard deviation to examination whether or not a population has a normal distribution. The formula to determine whether or not a population is ordinarily distributed are:
Q_one – (σ z_ane) + X
Q₃ – (σ z₃) + X
Where Q₁ is the commencement quartile, Q₃ is the third quartile, σ is the standard divergence, z is the standard score ("z-score") and X is the mean. In order to tell whether a population is ordinarily distributed, solve both equations so compare the results. If there is a meaning difference between the results and the first or third quartiles, so the population is not normally distributed.

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What is an Interquartile Range Used For?

The IQR is used to measure how spread out the information points in a set are from the mean of the data set. The higher the IQR, the more spread out the data points; in contrast, the smaller the IQR, the more bunched upwardly the data points are around the hateful. The IQR range is one of many measurements used to mensurate how spread out the data points in a data set are. It is best used with other measurements such as the median and full range to build a consummate picture show of a data set's trend to cluster effectually its mean.
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Where Does the term Interquartile Range Come From?

Who invented the term "Interquartile Range?" In order to find that out, we have to go dorsum to the 19th century.

History

British dr. Sir Donald MacAlister used the terms lower quartile and college quartile in the 1879 publication, the Law of the Geometric Hateful. 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."
origin of the word quartile

Although a physician past trade, he was gifted with mathematics and achieved the highest score in the last mathematics exams at Cambridge University in 1877. He spoke xix 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). Nevertheless, it wasn't until 1882 that Galton ("Written report of the Anthropometric Commission") used the upper quartile and lower quartile values and the term "interquartile range" —defined as twice the probable error. Galton wasn't but a statistician—he was also an anthropologist, geographer, proto-genetecist and psychometrician who produced more than 340 books. He as well coined the statistical terms "correlation" and "regression toward the mean."

References

Gonick, L. and Smith, West. The Cartoon Guide to Statistics. New York: Harper Perennial, pp. xx-21, 1993.

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