A data set has an even number of values. How do you find the median?

Put the values in size order, then find the midpoint of the middle two . Add those two values and divide by 2, so the median of 1, 2, 3, 4 is 2.5.

A data set contains one extremely large value. Which average is affected most by it, and why?

The mean is affected most, because it is worked out from every value in the set. The median depends only on the value or values in the middle, so an extreme value barely moves it.

Which average can you use for non-numerical data such as dog, cat, cat, fish?

Only the mode can be used, because it is simply the value that appears most often. The mean and the median both need data that can be added up or put in size order.

True or False? The range is affected by outliers but the interquartile range is not.

True. The range is the largest value minus the smallest value, so a single extreme value changes it completely. The interquartile range is worked out from the two quartiles only, so values beyond them have no effect on it at all.

For the data set 1, 2, 2, 5, 5, 6, why would the mode be a poor choice of average?

This set has two modes , 2 and 5, so the mode does not give a single clear average. A data set can also have no mode at all , when no value is repeated.

Define standard deviation .

The standard deviation measures how spread out a set of data is about its mean . A larger standard deviation means the values are more spread out, and it is measured in the same units as the data.

A data set contains an outlier. Should you remove it before calculating?

Removing an outlier depends on the context, so the value has to be checked first. Remove it if it turns out to be an error , such as 17 recorded as 71, but keep it if it is a valid part of the sample, such as the director's salary among a company's salaries.

A mean number of people works out as 7.5. Is that a problem?

No, a mean does not have to be a whole number, even when the data being averaged are counts. It is calculated from the whole data set, so it need not be a value that could actually occur.

What is the difference between a measure of location and a measure of dispersion?

A measure of location says where the data sits, such as the median or a quartile. A measure of dispersion says how spread out it is, such as the range, the interquartile range or the standard deviation.

Level 3MathsAQAMathematical Studies (Core Maths)Paper 1: Data, Finance, Estimation & ModellingFlashcardsAnalysis of DataRepresenting Data Numerically

Representing Data Numerically (AQA Level 3 Mathematical Studies (Core Maths): Paper 1: Data, Finance, Estimation & Modelling): Flashcards

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FrontRepresenting Data Numerically
A data set has an even number of values. How do you find the median?

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A data set has an even number of values. How do you find the median?
Put the values in size order, then find the midpoint of the middle two.
Add those two values and divide by 2, so the median of 1, 2, 3, 4 is 2.5.
Complete the formula for the mean of data given in a frequency table.
$\bar{x} = \frac{\sum______}{\sum______}$
The completed formula is:
$\bar{x} = \frac{\sum f x}{\sum f}$
Here $\sum f x$ is the total of each value multiplied by its frequency, and $\sum f$ is the total frequency.
For grouped data, use the mid-interval value of each class as the value of $x$ .
A data set contains one extremely large value. Which average is affected most by it, and why?
The mean is affected most, because it is worked out from every value in the set.
The median depends only on the value or values in the middle, so an extreme value barely moves it.
Define interquartile range.
The interquartile range is the spread of the middle 50% of the data, found by subtracting the lower quartile from the upper quartile.
$I Q R = Q_{3} - Q_{1}$
It is measured in the same units as the data.
Which average can you use for non-numerical data such as dog, cat, cat, fish?
Only the mode can be used, because it is simply the value that appears most often.
The mean and the median both need data that can be added up or put in size order.
A data set of $n$ values is written in size order. Complete the positions of the lower quartile, the median and the upper quartile.
$Q_{1} is at position \frac{n + 1}{______}$
$Q_{2} is at position \frac{n + 1}{______}$
$Q_{3} is at position \frac{______(n + 1)}{4}$
The completed positions are:
$Q_{1} is at position \frac{n + 1}{4}$
$Q_{2} is at position \frac{n + 1}{2}$
$Q_{3} is at position \frac{3 (n + 1)}{4}$
For a large data set the same positions are used with $n$ replacing $n + 1$ throughout, and either version can be used.
True or False?
The range is affected by outliers but the interquartile range is not.
True.
The range is the largest value minus the smallest value, so a single extreme value changes it completely.
The interquartile range is worked out from the two quartiles only, so values beyond them have no effect on it at all.
For the data set 1, 2, 2, 5, 5, 6, why would the mode be a poor choice of average?
This set has two modes, 2 and 5, so the mode does not give a single clear average.
A data set can also have no mode at all, when no value is repeated.
How far from the nearest quartile must a value be to count as an outlier?
A value is an outlier when it lies more than $1.5 \times I Q R$ beyond the nearest quartile.
$x < Q_{1} - 1.5 \times I Q R or x > Q_{3} + 1.5 \times I Q R$
Define standard deviation.
The standard deviation measures how spread out a set of data is about its mean.
A larger standard deviation means the values are more spread out, and it is measured in the same units as the data.
A data set contains an outlier. Should you remove it before calculating?
Removing an outlier depends on the context, so the value has to be checked first.
Remove it if it turns out to be an error, such as 17 recorded as 71, but keep it if it is a valid part of the sample, such as the director's salary among a company's salaries.
Complete the formula for the standard deviation of a sample.
$σ_{n - 1} = \sqrt{\frac{\sum {(x -______)}^{2}}{______}}$
The completed formula is:
$σ_{n - 1} = \sqrt{\frac{\sum {(x - \bar{x})}^{2}}{n - 1}}$
Each value's distance from the mean is squared, and the total is divided by $n - 1$ rather than by $n$ because this is the standard deviation of a sample.
For the same set of data, is the sample standard deviation larger or smaller than the population one?
The sample standard deviation is the larger of the two, because it divides by $n - 1$ instead of by $n$ .
Dividing by a smaller number gives a bigger result, which is why the two options on a calculator do not agree.
A mean number of people works out as 7.5. Is that a problem?
No, a mean does not have to be a whole number, even when the data being averaged are counts.
It is calculated from the whole data set, so it need not be a value that could actually occur.
What is the difference between a measure of location and a measure of dispersion?
A measure of location says where the data sits, such as the median or a quartile.
A measure of dispersion says how spread out it is, such as the range, the interquartile range or the standard deviation.

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