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Measures of location and spread

Measures of spread: Range

Measures of spread: Range

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Tutor: Labib

Summary

Measures of spread: Range

In a nutshell

Measures of spread, also called measures of dispersion or variation, describe how spread out the data is.



Range

The range is the difference between the maximum and the minimum values of the data set.



Interquartile range

The interquartile range is the difference between the upper and the lower quartile.


​IQR=Q3−Q1{IQR=Q_3-Q_1}IQR=Q3​−Q1​​​

​IQRIQRIQR​​

Interquartile range

​Q3Q_3Q3​​​

Upper quartile

​Q1Q_1Q1​​​

Lower quartile



Interpercentile range

The interpercentile range is the difference between the values of two chosen percentiles.


Note: The range is affected by extreme values, whereas the interquartile and interpercentile ranges aren't. However, the range takes into account all of the data, while the interquartile range only considers the middle 50%50\%50%.


Example 1

For the following values, find the range, the interquartile range and the 10th10th10th to 90th90th90th interpercentile range.


3,5,5,7,9,10,12,12,15,16,20,20,213,5,5,7,9,10,12,12,15,16,20,20,213,5,5,7,9,10,12,12,15,16,20,20,21​​


The range will be given by:


Maximum−minimum=21−3=18‾Maximum-minimum=21-3=\underline{18}Maximum−minimum=21−3=18​​​


For the interquartile range you have to find each quartile:


​Q1=134=3.25=4thQ_1=\dfrac{13}{4}=3.25=4thQ1​=413​=3.25=4th value (rounding up).


Q3=13×34=9.75=10thQ_3=\dfrac{13\times3}{4}=9.75=10thQ3​=413×3​=9.75=10th​ value.

​

So, the lower quartile will be 777 and the upper quartile is 161616.


Q3−Q1=16−7=9‾Q_3-Q_1=16-7=\underline{9}Q3​−Q1​=16−7=9​​​


Now, find the 10th10th10th and the 90th90th90th percentiles:


0.1×13=1.3≈2nd0.1\times 13=1.3\approx2nd0.1×13=1.3≈2nd value (rounding up), which means P10=5P_{10}=5P10​=5.

0.9×13=11.7≈12th0.9\times13=11.7\approx 12th0.9×13=11.7≈12th value, which means P90=20P_{90}=20P90​=20.


P90−P10=20−5=15‾P_{90}-P_{10}=20-5=\underline{15}P90​−P10​=20−5=15​​​​


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Learn with Basics

Learn the basics with theory units and practise what you learned with exercise sets!
Length:
Measures of central tendency: Mean, median, mode

Unit 1

Measures of central tendency: Mean, median, mode

Measures of location: Quartiles, percentiles, deciles

Unit 2

Measures of location: Quartiles, percentiles, deciles

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Score 80% to jump directly to the final unit.

Optional

Measures of spread: Range

Unit 3

Measures of spread: Range

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FAQs - Frequently Asked Questions

What is the interquartile range?

It is the difference between the upper and the lower quartile.

What are examples of measures of spread?

The range, the interquartile range and the interpercentile range.

What is a measure of spread?

A measure of spread describes how spread out the data values in a set are.

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