Range and Quartiles
In this article we shall explore range and quartiles. Below is a quick summary of what is covered in this article.
Summary
- Range of set of data is the difference between the highest and the lowest value in the set of data or observations.
- Quartiles, Q1, Q2, Q3, Q4 split the data into four parts
- For discreete data to find the lower quartile Q1, divide n by 4. To calculate the upper quartile Q3 divide n by 4 and multiply 3. When the result is a whole number find the mid-point of the corresponding term and the term above. When the result is not a whole number round the number up and pick the corresponding term.
- For continuous data find the lower quartile Q1, divide n by 4 and to find the upper quartile Q3 divide n by 4 and multiply 3 and then use interpolation to find the value of the corresponding term.
- The interquartile range is the upper quartile Q3 – lower quartile Q1
Range
To find the range of a set of data we simply find the difference between the highest and lowest value.
Quartiles
Data can be split into 4 parts Q1, Q2, Q3 as shown in the figure below;
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There are many ways of finding the quartiles which may all provide different results. In this article we shall divide the number of observations n by 4 and use interpolation to find the corresponding value.
To calculate the upper quartile Q3, we shall divide the number of observations by 4 and multiply by three and then use interpolation to find the value in the position.
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Range
Range = highest value – lowest value
Range = 16 – 2 = 14
Interquartile
To find the interquartile we must find the lower and upper quartile first. We know that;
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…where n is the number of observations.
In the question there are 11 observations, therefore;
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…so we must round up which gives 4. The lower bound Q1 is therefore the third term when the numbers are put in ascending order.
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Next we must find the upper quartile.
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…where n is the number of observations.
There are 11 observations in the question
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Rounding up 8.25 gives 9. The upper quartile Q3 is therefore the ninth value when the numbers are put in ascending order.
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The interquartile is the difference between the upper and lower quartile;
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Simon records the number of music CDs in the collection of students in her year. The results are shown in the table;
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Find the interquartile.
We must first calculate the cumulative frequency.
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Lower quartile
We can now use the culative frequency to find the lower quartile.
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Therefore the lower quartile Q1 is the 25th term which is 27.
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Upper quartile
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The upper quartile is the 75th term which is equal to 28.
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Interquartile
Therefore the interquartile;
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The length of time (to the nearest minutes) spent on the internet each evening by a group of students is shown in the table below. Calculate interquartile range.
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Lower quartile
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To find the value of the lower quartile we must use interpolation
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Upper quartile
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We must use interpolation
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Interquartile range
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