Which Method Can Be Used To Find The Interquartile Range For A Set Of Data?

What is an Interquartile Range?

The interquartile range is a measure of where the "eye 50" is in a data gear up. Where a range is a measure of where the outset and end are in a set, an interquartile range is a measure out of where the majority of the values lie. That'due south why information technology's preferred over many other measures of spread when reporting things like schoolhouse performance or Sabbatum scores.

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

IQR = Q_iii – Q₁.

Watch the video for how to calculate the interquartile range by hand:

How to find an interquartile range

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Solve the formula by hand.

Steps:

Step i: Put the numbers in order.
1, ii, 5, 6, vii, 9, 12, fifteen, 18, 19, 27.
Step ii: Detect the median.
one, 2, 5, 6, vii, ix, 12, xv, xviii, 19, 27.
Footstep 3: Identify parentheses around the numbers above and below the median.
Not necessary statistically, but it makes Q1 and Q3 easier to spot.
(1, 2, 5, 6, 7), 9, (12, fifteen, xviii, 19, 27).
Step 4: Find Q1 and Q3
Recollect of Q1 every bit a median in the lower half of the data and think of Q3 every bit a median for the upper one-half of data.
(1, 2, 5, 6, 7), 9, ( 12, 15, eighteen, nineteen, 27). Q1 = v and Q3 = 18.
Pace 5: Subtract Q1 from Q3 to find the interquartile range.
18 – 5 = 13.

Similar the caption? Cheque out the Practically Cheating Statistics Handbook, which has hundreds more step-by-step explanations, only like this one!

What if I Have an Even Set up of Numbers?

Example question: Find the IQR for the following data set: iii, v, vii, 8, ix, eleven, 15, xvi, 20, 21.

Step one: Put the numbers in order.
three, 5, seven, viii, ix, 11, 15, 16, twenty, 21.
Step 2: Brand a mark in the center of the information:
3, v, 7, 8, 9, | 11, 15, xvi, 20, 21.
Stride 3: Identify parentheses around the numbers above and below the mark you lot made in Step two--information technology makes Q1 and Q3 easier to spot.
(three, 5, 7, 8, 9), | (11, 15, 16, twenty, 21).
Stride iv: Find Q1 and Q3
Q1 is the median (the middle) of the lower half of the data, and Q3 is the median (the middle) of the upper one-half of the data.
(3, 5, seven, viii, 9), | (eleven, 15, 16, 20, 21). Q1 = 7 and Q3 = 16.
Pace 5: Decrease Q1 from Q3.
16 – seven = ix.
This is your IQR.

Discover an interquartile range for an odd set up of numbers: Alternate Method

As yous may already know, nothing is "ready 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 instance, in the following set of numbers: 1, 2, 5, half-dozen, seven, 9, 12, xv, 18, 19, 27 some statisticians would suspension it into 2 halves, including the median (9) in both halves:
(one, 2, 5, 6, 7, 9), (9, 12, 15, xviii, 19, 27)
This leads to two halves with an even prepare of numbers, then you can follow the steps above to detect the IQR.
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Box Plot interquartile range: How to observe it

Lookout the video for the steps.

How to discover an interquartile range on a boxplot

Tin't see the video? Click hither.

Box Plot interquartile range: How to find it

box plot interquartile range

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

Pace 1: Find Q1.Q1 is represented past the left hand border of the "box" (at the point where the whisker stops).

In the to a higher place graph, Q1 is approximately at 2.vi. (A complete caption of Q1 is here: The five number summary.)
Step 2: Find Q3.
Q3 is represented on a boxplot past the right paw edge of the "box".

Q3 is approximately 12 in this graph.
Step 3: Subtract the number you lot found in step i from the number you lot constitute in step three.
This will requite you the interquartile range. 12 – 2.vi = 9.four.

That's it!

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Interquartile Range in Minitab

Watch the video for step-by-step directions:

How to find an interquartile range in Minitab

Can't see the video? Click here.

Interquartile Range in Minitab: Steps

Instance question: Detect an interquartile range in Minitab for the Grade Point Average (GPA) in the post-obit information set:
Course Indicate Average (GPA): 1(3.ii), 1(3.1), 2(iii.5), 2(2.0), 3(1.nine), iii(4.0), 3(3.9), 4(3.eight), 4(ii.9), five(iii.ix), 5(3.2), 5(three.3), 6(three.four), 6(ii.6), half dozen(2.five), 7(2.0), 7(1.5), 8(4.0), eight(2.0).

Footstep 1: Blazon your data into a Minitab worksheet. Enter your data into one or ii columns.
minitab interquartile range a

Step two: Click "Stat," then click "Basic Statistics," then click "Brandish Descriptive Statistics" to open the Descriptive Statistics menu.
minitab interquartile range b

Step 3: Click a variable proper 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" push.
minitab interquartile range c

Stride 5: Check "Interquartile Range."
minitab interquartile range d .

Step 6: Click the "OK" button (a new window will open with the effect). The IQR for the GPA in this detail data fix is one.8.
minitab interquartile range e

That's it!

Tip: If you don't see descriptive statistics evidence in a window, click "Window" on the toolbar, and so click "Tile." Click the Session window (this is where descriptive statistics appear) and and so scroll up to see your results.

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Interquartile Range in Excel 2007

How to Observe an Interquartile Range Excel 2007

Picket 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 1: Enter your information into a single Excel column on a worksheet. For example, blazon your information in cells A2 to A10. Don't leave any gaps in your data.

Step 2: Click a blank cell (for example, click cell B2) and then type =QUARTILE(A2:A10,i). You'll need to supercede A2:A10 with the actual values from your data set. For example, if y'all typed your data into B2 to B50, the equation is =QUARTILE(B2:B50,ane). The "1" in this Excel formula(A2:A10,1) represents the beginning quartile (i.east the point lying at 25% of the data set up).

Step iii: Click a second blank cell (for case, click jail cell B3) and and so type =QUARTILE(A2:A10,3). Replace A2:A10 with the bodily 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 information ready).

Step 4: Click a third blank cell (for example, click cell B4) and and then type =B3-B2. If your quartile functions from Step 2 and 3 are in different locations, modify the cell references.

Pace 5: Press the "Enter" key. Excel will render the IQR in the cell yous clicked in Step 4

That's it!

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How to Find an Interquartile Range in SPSS

Like most technology, SPSS has several ways that you can calculate the IQR. Yet, if you click on the about intuitive way you would wait to find it ("Descriptive Statistics > Frequencies"), the surprise is that it won't list the IQR (although information technology will listing the first, 2d and third quartiles). You could take this route and and so subtract the tertiary quartile from the commencement to get the IQR. However, the easiest mode to find the interquartile range in SPSS by using the "Explore" control. If you accept already typed data into your worksheet, skip to Stride 3.

Watch the video for the steps:

How to observe the Interquartile Range in SPSS

Can't run into the steps? Click here.

Steps

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

Pace 2: Blazon your data into columns in the worksheet. You can utilize as many columns as you need, but don't leave blank rows or spaces between your data. Encounter: How to Enter Data into SPSS.

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

Step four: Click the variable name (that'due south but a fancy proper noun for the column heading), then click the top arrow to motility the variable into the "Dependent listing" box.

The "Explore" variables dialog box.

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

Tip: This example has only i listing typed into the data sheet, but y'all may have several to choose from depending on how y'all entered your information. Brand certain yous select the right variable (column names) before proceeding. If you want more memorable variable names, change the column title past clicking the "variable view" button at the very bottom left of the worksheet. Blazon in your new variable name and then render to information view past clicking the "information view" push.
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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 accept three, 7 and 28 in your set of data, imagine them as points on a number line that is centered on 0 simply stretches both infinitely beneath zero and infinitely higher up zero. Once plotted on that number line, the smallest data point and the biggest data point in the set up of information create the boundaries (i.e. a lower leap and an upper bound) of an interval of space on the number line that contains all data points in the prepare. The interquartile range (IQR) is the length of the middle 50% of that interval of infinite.

what is an interquartile range

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

If yous desire to know that the IQR is in formal terms, the IQR is calculated equally: The difference between the third or upper quartile and the first or lower quartile. Quartile is a term used to draw how to divide the set of information into iv equal portions (think quarter).

IQR Example

If you accept a set containing the data points 1, iii, 5, seven, viii, x, 11 and 13, the first quartile is 4, the 2nd quartile is seven.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 xiii. Every bit such, the IQR of that data gear up is half dozen.5, calculated equally 10.5 minus iv. The first and 3rd quartiles are also sometimes called the 25th and 75th percentiles considering those are the equivalent figures when the data set is divided into percents rather than quarters.

Interquartile Range using the TI83

Watch the video for the steps:

TI 83 Interquartile Range

Can't meet the video? Click hither.

While you tin use the nifty online interquartile range computer on this website, that might non be an choice in a quiz or exam. Most instructors allow the utilize of a TI-83 on tests, and it'south even one of the few calculators allowed in the AP Statistics exam. Finding the TI 83 interquartile range involves zip more than inbound your information list and pushing a couple of buttons.

Example problem: Find the TI 83 interquartile range for the heights of the top ten buildings in the world (as of 2009). The heights, (in feet) are: 2717, 2063, 2001, 1815, 1516, 1503, 1482, 1377, 1312, 1272.

Steps

Pace 1: Enter the above data into a listing on the TI 83 calculator. Printing the STAT button and then press ENTER. Enter the kickoff number (2717), and and so press ENTER. Continue entering numbers, pressing ENTER later on each entry.

Step two: Press the STAT button.

Step three: Press the right arrow button (the arrow keys are located at the acme right of the keypad) to select "Calc."

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

Step v: Printing ENTER once more to bring up a list of stats.

Step half dozen: Whorl down the listing with the arrow keys to find Q1 and Q3. Write those numbers down. Yous could copy and paste the numbers but unfortunately, Texas Instruments doesn't make this easy:

Use the arrow keys to identify the cursor at the beginning of the
text that you want to highlight.
Using the TI Keyboard, press and hold down the Shift key, and and then use the arrow keys to highlight the text.
Release the Shift key and pointer cardinal.

The copy and paste menu should appear, enabling y'all to copy and paste the information. You would have to do this twice (returning to the Home screen each time), and so it'southward much faster but to write the numbers down.

Pace 7:Subtract Q1 from Q3 to find the IQR (potent>624 feet for this fix of numbers).

That'due south information technology!
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How to Find Q1, Q3 and the Interquartile Range TI 89

Instance trouble: Notice Q₁, Q₃, and the IQR for the following listing of numbers: i, 9, 2, 3, 7, 8, 9, two.

Stride 1: Press APPS. Ringlet to Stats/Listing Editor (apply the arrow keys on the keypad to coil). Press ENTER. If you don't have the stats/list editor you tin can download it here.

Step ii: Clear the list editor of data: press F1 8.

Step 3: Printing ALPHA 9 Blastoff 1 ENTER. This names your list "IQ."

Step iv: Enter your numbers, one at a time. Follow each entry by pressing the ENTER key. For our group of numbers, enter
1,9,2,three,7,8,9,ii

Pace 5: Press F4, so ENTER (for the ane-var stats screen).

Step 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 in that location for you. Press ENTER twice.

Stride 7: Read the results. Q₁ is listed as Q1X (in our example, Q1X=2). Q₃ is listed as Q3X (Q3X=viii.five). To find the IQR, subtract Q_one from Q₃ on the Home screen. The IQR is eight.5-two=6.5.

That'south 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_one is the lower quartile.

the interquartile range formula

IQR as a test for normal distribution

Utilize 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:
Q_one – (σ z₁) + X
Q_three – (σ z₃) + X
Where Q₁ is the offset quartile, Q₃ is the third quartile, σ is the standard deviation, z is the standard score ("z-score") and 10 is the mean. In gild to tell whether a population is normally distributed, solve both equations and so compare the results. If in that location is a significant difference between the results and the starting time or tertiary quartiles, then the population is not normally distributed.

What is an Interquartile Range Used For?

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

Who invented the term "Interquartile Range?" In club to observe that out, nosotros take to get dorsum to the 19th century.

History

British physician Sir Donald MacAlister used the terms lower quartile and higher quartile in the 1879 publication, the Law of the Geometric Hateful. Proc. R. Soc. XXIX, p. 374: " "Every bit these 2 measures, with the hateful, separate the curve of facility into four equal parts, I suggest to telephone call them the 'higher quartile' and the 'lower quartile' respectively."
origin of the word quartile

Although a dr. by merchandise, he was gifted with mathematics and achieved the highest score in the concluding 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 really in response to a question put forward by Francis Galton (inventor of the Galton board). Still, 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 fault. Galton wasn't but a statistician—he was also an anthropologist, geographer, proto-genetecist and psychometrician who produced more than 340 books. He likewise coined the statistical terms "correlation" and "regression toward the hateful."

References

Gonick, 50. and Smith, W. The Cartoon Guide to Statistics. New York: Harper Perennial, pp. 20-21, 1993.

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