How To Find Interquartile Range For Grouped Data

What is an Interquartile Range?

The interquartile range is a measure of where the "middle l" 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 majority 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 get-go quartile subtracted from the third quartile:

IQR = Q₃ – Q₁.

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

How to discover an interquartile range

Can't see the video? Click hither.

Contents:

Solving by hand:

1. Solve the formula by hand (odd set of numbers).
2. What if I take an even set of numbers?
3. Discover an interquartile range for an odd set of numbers: Second Method
4. Box Plot interquartile range: How to find it

Using Technology:

1. Interquartile Range in Minitab
2. Interquartile Range in Excel
3. Interquartile Range in SPSS
4. Interquartile Range on the TI83
5. Q1, Q3 and the IQR on the TI89

Full general info:

1. What is an Interquartile range?
2. What is the Interquartile Range Formula?
3. IQR equally a Test for Normal Distribution
4. What is an Interquartile Range used for?
5. History of the Interquartile Range.

Solve the formula by paw.

Steps:

• Step ane: Put the numbers in order.
  1, 2, v, 6, seven, 9, 12, fifteen, xviii, nineteen, 27.
• Step 2: Find the median.
  1, 2, five, 6, vii, nine, 12, fifteen, xviii, nineteen, 27.
• Stride 3: Place parentheses around the numbers above and below the median.
  Not necessary statistically, merely it makes Q1 and Q3 easier to spot.
  (1, ii, five, vi, 7), 9, (12, fifteen, 18, 19, 27).
• Step 4: Find Q1 and Q3
  Remember of Q1 as a median in the lower one-half of the data and think of Q3 equally a median for the upper half of data.
  (1, 2, 5, six, 7),  9, ( 12, xv, 18, xix, 27). Q1 = 5 and Q3 = 18.
• Step 5: Subtract Q1 from Q3 to observe the interquartile range.
  18 – five = 13.

Like the caption? Check out the Practically Adulterous Statistics Handbook, which has hundreds more step-by-pace explanations, just like this one!

What if I Take an Even Set of Numbers?

Example question: Detect the IQR for the following information set: three, v, vii, 8, ix, xi, fifteen, xvi, 20, 21.

• Step 1: Put the numbers in order.
  3, 5, vii, eight, 9, xi, xv, 16, xx, 21.
• Step two: Brand a mark in the center of the information:
  3, 5, vii, 8, 9, | 11, 15, sixteen, 20, 21.
• Step 3: Place parentheses effectually the numbers above and below the marker you made in Step 2--information technology makes Q1 and Q3 easier to spot.
  (3, 5, 7, 8, 9), | (11, 15, 16, twenty, 21).
• Stride 4: Notice 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 half of the data.
  (iii, 5, 7, viii, 9), | (11, 15, sixteen, 20, 21). Q1 = 7 and Q3 = 16.
• Step five: Subtract Q1 from Q3.
  16 – seven = 9.
  This is your IQR.

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

Equally yous may already know, nix is "set in stone" in statistics: when some statisticians discover an interquartile range for a prepare of odd numbers, they include the median in both both quartiles. For case, 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 (nine) in both halves:
(1, 2, 5, 6, 7, 9), (ix, 12, xv, 18, nineteen, 27)
This leads to two halves with an even set of numbers, so y'all tin 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 discover an interquartile range on a boxplot

Can't see the video? Click here.

Box Plot interquartile range: How to find it

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

• Step 1: Notice Q1.Q1 is represented by the left mitt edge of the "box" (at the point where the whisker stops).
  

  In the to a higher place graph, Q1 is approximately at ii.6. (A complete explanation of Q1 is here: The five number summary.)

• Footstep 2: Notice Q3.
  Q3 is represented on a boxplot by the right hand edge of the "box".
  
  Q3 is approximately 12 in this graph.
• Pace 3: Decrease the number you found in step 1 from the number you found in footstep 3.
  This will give yous the interquartile range. 12 – 2.six = nine.4.

That'south it!

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Interquartile Range in Minitab

Lookout the video for step-past-footstep directions:

How to find an interquartile range in Minitab

Can't see the video? Click here.

Interquartile Range in Minitab: Steps

Example question: Find an interquartile range in Minitab for the Class Point Average (GPA) in the following data set:
Grade Betoken Average (GPA): one(3.2), ane(3.ane), 2(iii.5), 2(two.0), 3(1.9), 3(4.0), iii(3.9), 4(3.8), iv(2.nine), 5(3.nine), 5(iii.2), five(3.3), vi(3.4), 6(2.6), 6(two.five), 7(2.0), seven(1.five), 8(4.0), 8(ii.0).

Step 1: Type your information into a Minitab worksheet. Enter your data into i or ii columns.

Stride 2: Click "Stat," then click "Bones Statistics," then click "Display Descriptive Statistics" to open the Descriptive Statistics menu.

Step 3: Click a variable proper noun in the left window then click the "Select" push to transfer the variable proper noun to the correct-hand window.

Stride 4: Click the "Statistics" button.

Step five: Check "Interquartile Range."
.

Step half-dozen: Click the "OK" push button (a new window volition open with the result). The IQR for the GPA in this particular data prepare is 1.8.

That'southward it!

Tip: If y'all 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 so curlicue up 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 find 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 example, type your data in cells A2 to A10. Don't leave whatever gaps in your information.

Step 2: Click a blank cell (for example, click jail cell B2) and then type =QUARTILE(A2:A10,1). You'll demand to replace A2:A10 with the actual values from your information gear up. For instance, if you typed your data into B2 to B50, the equation is =QUARTILE(B2:B50,i). The "one" in this Excel formula(A2:A10,i) represents the starting time quartile (i.e the point lying at 25% of the data set).

Step 3: Click a second bare prison cell (for case, click cell B3) and then type =QUARTILE(A2:A10,3). Supplant A2:A10 with the bodily values from your information set. The "3" in this Excel formula (A2:A10,3) represents the third quartile (i.due east. the point lying at 75% of the data prepare).

Footstep iv: Click a third blank cell (for instance, click jail cell B4) and and then blazon =B3-B2. If your quartile functions from Footstep 2 and 3 are in different locations, modify the prison cell references.

Stride v: Press the "Enter" key. Excel will return the IQR in the cell you clicked in Step 4

That'due south information technology!

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How to Detect an Interquartile Range in SPSS

Like near engineering, SPSS has several ways that you can summate the IQR. However, if you click on the near intuitive way you would await to discover it ("Descriptive Statistics > Frequencies"), the surprise is that it won't list the IQR (although it volition list the kickoff, second and third quartiles). You could have this route and then subtract the third quartile from the first to get the IQR. Even so, the easiest way to find the interquartile range in SPSS by using the "Explore" control. If yous have already typed data into your worksheet, skip to Step iii.

Watch the video for the steps:

How to find the Interquartile Range in SPSS

Tin't see the steps? Click here.

Steps

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

Step two: Type your data into columns in the worksheet. Yous can utilize as many columns as y'all need, only don't leave bare rows or spaces betwixt your data. See: How to Enter Data into SPSS.

Step three: Click "Analyze," then mouse over "Descriptive Statistics." Click "Explore" to open the "Explore" dialog box.

Pace four: Click the variable name (that'south just a fancy name for the cavalcade heading), then click the superlative arrow to movement the variable into the "Dependent list" box.

The "Explore" variables dialog box.

Step 5: Click "OK." The interquartile range is listed in the Descriptives box.

Tip: This case has only one list typed into the data sail, but you may take 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, alter the cavalcade title past clicking the "variable view" button at the very lesser left of the worksheet. Type in your new variable name and then render to data 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 information, imagine them equally points on a number line that is centered on 0 only stretches both infinitely below null and infinitely in a higher place zip. Once plotted on that number line, the smallest data betoken and the biggest data point in the set of data create the boundaries (i.east. a lower leap and an upper bound) of an interval of space on the number line that contains all data points in the set. The interquartile range (IQR) is the length of the centre 50% of that interval of space.

The interquartile range is the middle fifty% of a data set. 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 third or upper quartile and the outset or lower quartile. Quartile is a term used to describe how to split up the gear up of data into iv equal portions (think quarter).

IQR Example

If you accept a prepare containing the data points 1, three, 5, seven, 8, ten, xi and 13, the first quartile is four, the 2nd quartile is seven.5 and the tertiary quartile is 10.five. Draw these points on a number line and you lot'll see that those three numbers divide the number line in quarters from 1 to 13. As such, the IQR of that data fix is vi.v, calculated equally 10.five minus four. The first and 3rd quartiles are also sometimes called the 25th and 75th percentiles considering those are the equivalent figures when the information set is divided into percents rather than quarters.

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Interquartile Range using the TI83

Watch the video for the steps:

TI 83 Interquartile Range

Can't come across the video? Click here.

While you can apply the nifty online interquartile range calculator on this website, that might not be an option in a quiz or test. Most instructors let the use of a TI-83 on tests, and it's even one of the few calculators allowed in the AP Statistics examination. Finding the TI 83 interquartile range involves aught 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 (every bit of 2009). The heights, (in feet) are: 2717, 2063, 2001, 1815, 1516, 1503, 1482, 1377, 1312, 1272.

Steps

Step 1: Enter the above data into a list on the TI 83 reckoner. Press the STAT button and then press ENTER. Enter the starting time number (2717), and then printing ENTER. Continue inbound numbers, pressing ENTER afterwards each entry.

Step 2: Printing the STAT button.

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

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

Stride 5: Printing ENTER again to bring upwards a list of stats.

Stride 6: Whorl downwardly the list with the arrow keys to find Q1 and Q3. Write those numbers down. You lot could copy and paste the numbers but unfortunately, Texas Instruments doesn't brand this like shooting fish in a barrel:

1. Use the arrow keys to place the cursor at the beginning of the
   text that yous want to highlight.
2. Using the TI Keyboard, press and concur down the Shift central, so use the arrow keys to highlight the text.
3. Release the Shift key and arrow fundamental.

The copy and paste menu should appear, enabling y'all to copy and paste the data. You would have to do this twice (returning to the Habitation screen each time), and then information technology's much faster just to write the numbers down.

Pace 7:Subtract Q1 from Q3 to discover the IQR (stiff>624 feet for this set of numbers).

That's it!
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How to Find Q1, Q3 and the Interquartile Range TI 89

Example problem: Notice Q₁, Q₃, and the IQR for the following list of numbers: one, 9, two, three, 7, viii, 9, ii.

Step 1: Press APPS. Scroll to Stats/Listing Editor (use the arrow keys on the keypad to ringlet). Press ENTER. If y'all don't have the stats/list editor y'all can download it here.

Footstep 2: Clear the list editor of data: printing F1 8.

Step 3: Printing Alpha ix ALPHA one ENTER. This names your list "IQ."

Step 4: Enter your numbers, one at a time. Follow each entry by pressing the ENTER cardinal. For our group of numbers, enter
one,9,2,3,seven,eight,9,two

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

Step 6: Tell the calculator you lot want stats for the list chosen "IQ" by entering Blastoff 9 ALPHA 1 into the "List:" box. The figurer should automatically put the cursor there for y'all. Press ENTER twice.

Stride 7: Read the results. Q₁ is listed every bit Q1X (in our case, Q1X=2). Q₃ is listed every bit Q3X (Q3X=8.v). To detect the IQR, subtract Q₁ from Q_iii on the Home screen. The IQR is 8.5-2=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 Q₁ is the lower quartile.

IQR as a test for normal distribution

Apply 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 non a population is commonly distributed are:
Q₁ – (σ z₁) + X
Q_three – (σ z₃) + X
Where Q_i is the first quartile, Q₃ is the third quartile, σ is the standard deviation, z is the standard score ("z-score") and X is the mean. In gild to tell whether a population is normally distributed, solve both equations and so compare the results. If there is a significant difference between the results and the showtime or third quartiles, then the population is not normally distributed.

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What is an Interquartile Range Used For?

The IQR is used to mensurate how spread out the data points in a ready are from the mean of the information set. The higher the IQR, the more spread out the data points; in dissimilarity, the smaller the IQR, the more bunched upwardly the data points are around the mean. The IQR range is one of many measurements used to measure how spread out the data points in a data set up are. Information technology is best used with other measurements such every bit the median and total range to build a consummate picture of a data fix'due south trend to cluster around its mean.
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Where Does the term Interquartile Range Come up From?

Who invented the term "Interquartile Range?" In order to notice that out, nosotros have to go 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 Mean. Proc. R. Soc. XXIX, p. 374: " "As these 2 measures, with the hateful, split up the curve of facility into four equal parts, I propose to phone call them the 'higher quartile' and the 'lower quartile' respectively."

Although a physician past trade, he was gifted with mathematics and accomplished the highest score in the concluding mathematics exams at Cambridge University in 1877. He spoke nineteen languages including English, Czech and Swedish.

Macalister's newspaper, the Police force of the Geometric Mean was actually in response to a question put forward by Francis Galton (inventor of the Galton board). Nonetheless, 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" —divers 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 than 340 books. He also 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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