What do the lower quartile and the upper quartile tell you about a data set?

The lower quartile has a quarter of the data below it, and the upper quartile has three quarters of the data below it. Together with the median, which is sometimes called the second quartile, they split an ordered data set into four equal parts.

How do you find the quartiles by splitting an ordered data set in half, and what changes if there is an odd number of values?

The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half. With an even number of values every value goes into one half or the other, but with an odd number the median itself is left out of both halves.

The lower quartile works out as the 3.25th value, the 3rd value is 12 and the 4th is 14. What is the lower quartile?

Go a quarter of the way from 12 to 14, which gives a lower quartile of 12.5. A position ending in .25 or .75 means the quartile lies a quarter or three quarters of the way between the two values on either side of it.

True or False? The two methods for finding quartiles always give the same answer.

False. For the same 20 values, splitting the data in half gives a lower quartile of 36.5, while the position formula gives 36.25. Both are legitimate ways of dividing an ordered data set into quarters, so neither answer is more correct than the other.

Define a percentile .

Percentiles divide a data set into 100 equal parts. The n th percentile has n % of the data values below it, so 90% of the values lie above the 10th percentile. Percentiles do not have to be whole numbers, and the 2.5th percentile is a perfectly sensible thing to talk about.

Why are percentiles useful for describing something like incomes?

Percentiles let you talk about particular slices of a distribution rather than only its centre. Comparing the highest 1% of earners, who sit above the 99th percentile, with the median income says far more than the median could on its own.

What does a measure of spread tell you that an average does not?

An average tells you what a typical value is, while a measure of spread tells you how scattered the data is around that typical value. A small range means the values cluster closely around the average, and a large one means some of them sit a long way from it.

Which part of a data set does the interquartile range measure the spread of?

The middle half of the data, lying between the lower quartile and the upper quartile. Half of all the values fall in that range, which can be thought of as the most typical half of the data set.

A data set of times is measured in seconds. What are the units of its range and of its interquartile range?

Both are in seconds , the same units as the data values themselves. Each one is a difference between two data values, so it carries whatever units those values had.

True or False? The range of a data set can be zero.

True. If every value in the data set is the same, then the largest and the smallest are equal and the difference between them is 0. A range of zero means there is no spread at all, which is a perfectly sensible thing for a data set to have.

An outlier is an extreme data value that does not fit the general pattern of the rest of the data. At this level they are found by inspection, which means looking down the data for values that are much bigger or much smaller than everything else.

What effect does a single outlier have on the mean, the median and the mode?

A single outlier can have a big effect on the mean , because every value in the data set goes into that calculation. The median is barely affected, because it depends on the position of the middle value rather than on how extreme the end values are. The mode is usually not affected at all, since one unusual value is unlikely to become the most common one.

Ages at which students sat a GCSE are 3, 13, 15, 15, 15, 15, 16, 16, 16, 16, 16 and 57. Which values are outliers?

The 3 and the 57 . Almost all of the data sits at 15 or 16, and 13 is only a little below that, but 3 and 57 are both a very long way from the rest of the values.

Ages at which students sat a GCSE include an outlier of 3 and an outlier of 57. Should both be removed from the data set?

No. The 3 is almost certainly a recording error, because a 3 year old could not sit a GCSE, so it should be removed. The 57 should be kept, because older people do sometimes sit GCSEs, so it may well be a genuine value.

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Measures of Dispersion (AQA GCSE Statistics: Foundation): Flashcards

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What do the lower quartile and the upper quartile tell you about a data set?

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What do the lower quartile and the upper quartile tell you about a data set?
The lower quartile has a quarter of the data below it, and the upper quartile has three quarters of the data below it.
Together with the median, which is sometimes called the second quartile, they split an ordered data set into four equal parts.
How do you find the quartiles by splitting an ordered data set in half, and what changes if there is an odd number of values?
The lower quartile is the median of the lower half, and the upper quartile is the median of the upper half.
With an even number of values every value goes into one half or the other, but with an odd number the median itself is left out of both halves.
Complete the quartile positions for a list of $n$ raw data values written in order.
$lower quartile at \frac{n + 1}{______}, upper quartile at______\times \frac{n + 1}{4}$
The completed positions are:
$lower quartile at \frac{n + 1}{4}, upper quartile at 3 \times \frac{n + 1}{4}$
For 11 values that gives the 3rd and the 9th value exactly, but most totals give a position that is not a whole number.
The lower quartile works out as the 3.25th value, the 3rd value is 12 and the 4th is 14. What is the lower quartile?
Go a quarter of the way from 12 to 14, which gives a lower quartile of 12.5.
A position ending in .25 or .75 means the quartile lies a quarter or three quarters of the way between the two values on either side of it.
True or False?
The two methods for finding quartiles always give the same answer.
False.
For the same 20 values, splitting the data in half gives a lower quartile of 36.5, while the position formula gives 36.25.
Both are legitimate ways of dividing an ordered data set into quarters, so neither answer is more correct than the other.
Define a percentile.
Percentiles divide a data set into 100 equal parts.
The nth percentile has n% of the data values below it, so 90% of the values lie above the 10th percentile.
Percentiles do not have to be whole numbers, and the 2.5th percentile is a perfectly sensible thing to talk about.
Complete the links between percentiles and the other measures of position.
The 25th percentile is the $______$ quartile, the 50th percentile is the $______$ , and the 75th percentile is the upper quartile.
The completed links are:
The 25th percentile is the lower quartile, the 50th percentile is the median, and the 75th percentile is the upper quartile.
48 students are timed, and the first two classes of a grouped frequency table hold 30 of them between them. Which class contains the 65th percentile?
The 65th percentile sits at position $48 \times \frac{65}{100} = 31.2$ , so it lies between the 31st and the 32nd values.
The first two classes account for values 1 to 30, so both the 31st and the 32nd fall in the third class.
Why are percentiles useful for describing something like incomes?
Percentiles let you talk about particular slices of a distribution rather than only its centre.
Comparing the highest 1% of earners, who sit above the 99th percentile, with the median income says far more than the median could on its own.
Complete the two measures of spread.
$range = largest value -______, IQR =______- lower quartile$
The completed measures are:
$range = largest value - smallest value, IQR = upper quartile - lower quartile$
What does a measure of spread tell you that an average does not?
An average tells you what a typical value is, while a measure of spread tells you how scattered the data is around that typical value.
A small range means the values cluster closely around the average, and a large one means some of them sit a long way from it.
Which part of a data set does the interquartile range measure the spread of?
The middle half of the data, lying between the lower quartile and the upper quartile.
Half of all the values fall in that range, which can be thought of as the most typical half of the data set.
Germination times run from 5 to 11 days apart from a single value of 23. Why is the interquartile range a better measure of spread here than the range?
The 23 makes the range $23 - 5 = 18$ days, which suggests the data is far more spread out than it really is.
The largest and smallest values have no effect at all on the interquartile range, so it still describes where the bulk of the data actually sits.
A data set of times is measured in seconds. What are the units of its range and of its interquartile range?
Both are in seconds, the same units as the data values themselves.
Each one is a difference between two data values, so it carries whatever units those values had.
True or False?
The range of a data set can be zero.
True.
If every value in the data set is the same, then the largest and the smallest are equal and the difference between them is 0.
A range of zero means there is no spread at all, which is a perfectly sensible thing for a data set to have.
Define an outlier.
An outlier is an extreme data value that does not fit the general pattern of the rest of the data.
At this level they are found by inspection, which means looking down the data for values that are much bigger or much smaller than everything else.
Complete the two reasons a data set can contain an outlier.
An outlier may be a genuine $______$ event, or it may be a $______$ made when the data was recorded.
The completed reasons are:
An outlier may be a genuine extreme event, or it may be a mistake made when the data was recorded.
What effect does a single outlier have on the mean, the median and the mode?
A single outlier can have a big effect on the mean, because every value in the data set goes into that calculation.
The median is barely affected, because it depends on the position of the middle value rather than on how extreme the end values are.
The mode is usually not affected at all, since one unusual value is unlikely to become the most common one.
Ages at which students sat a GCSE are 3, 13, 15, 15, 15, 15, 16, 16, 16, 16, 16 and 57. Which values are outliers?
The 3 and the 57.
Almost all of the data sits at 15 or 16, and 13 is only a little below that, but 3 and 57 are both a very long way from the rest of the values.
Ages at which students sat a GCSE include an outlier of 3 and an outlier of 57. Should both be removed from the data set?
No.
The 3 is almost certainly a recording error, because a 3 year old could not sit a GCSE, so it should be removed.
The 57 should be kept, because older people do sometimes sit GCSEs, so it may well be a genuine value.

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