An ice cream seller records how many of each flavour were sold. Why is the mode the only average that can be used on this data?

Because the data values are the flavours , which are qualitative, and there is no way to add up or order a list of names. The numbers sold are the frequencies rather than the data values, which is what makes this data easy to mistake for numerical data.

Ten employees earn between £256 and £458 except one who earns £3850. The median wage is £401 and the mean is £722.60. Which average describes the wages better?

The median , because nine of the ten employees earn far less than the mean does. The single very large wage drags the mean upwards until it no longer describes a typical employee at all, while the median is left untouched by it.

Why is the mean unreliable when a grouped data set has an open-ended class?

Because an open-ended class has no upper boundary, so there is no midpoint to put into the calculation. Any value chosen for it is a guess, and since the mean depends on every class the whole estimate becomes unreliable.

True or False? The median can be used to work out other statistics, such as the quartiles and the skew.

True. The median is the starting point for the quartiles and therefore for the interquartile range, and it is used again when deciding how a distribution is skewed. The mode, by contrast, cannot be used to work out an associated measure of spread at all.

When comparing two data sets, why must you use the same average for both?

Because a mean and a median measure different things, so setting one against the other says nothing about how the two data sets actually differ. Only the same average, worked out the same way for both sets, gives a fair comparison.

True or False? Two people comparing the same pair of data sets must reach the same conclusion.

False. The conclusion depends on which average is chosen, and different averages can point in different directions when the data is skewed. Someone with an argument to make, such as a politician, may pick whichever average supports it best.

What two things must you compare when comparing two data sets?

An average , which is a measure of central tendency, and a measure of spread . One on its own is never enough, because two data sets can share the same average and still be spread out completely differently.

Why should the interquartile range not be used alongside the mean?

Because the mean uses every value in the data set, while the interquartile range deliberately ignores the top and bottom quarters of it. The two describe different parts of the data, so the range, which also depends on the extreme values, is what pairs with the mean.

What four things should a written comparison of two data sets contain?

A comparison of the two averages as numbers, followed by what that difference means in the context of the question. Then a comparison of the two measures of spread as numbers, followed by what that difference means in context as well. The numbers alone do not answer the question, so each one has to be turned back into a statement about the real situation.

Give three different ways of saying that one data set has a smaller spread than another.

Its values are less spread out , or they are closer together . You can also say that the values are more consistent , or that there is less variation between them.

Why might a conclusion drawn from comparing two data sets still be unreliable?

The data sets may be too small to represent what they are meant to, since five pupils' heights say nothing about a whole school. They may also be biased , for example if only the older year groups were measured, which would push the average height up.

Why might you compare data from a sample with data from the whole population?

To judge how representative the sample is. If the sample's average and spread come out close to the population's, it is a sound basis for drawing conclusions, and if they are far apart it is not.

True or False? A teacher and an examiner might want opposite things from the spread of a set of marks.

True. A teacher may want a small spread, because it suggests the whole class has understood the topic to a similar standard. An examiner may want a large spread, because it makes the grade boundaries easier to place.

Skewness describes the way the data in a distribution leans, rather than where its centre lies. A distribution can be skewed towards one side or the other, or it can be symmetrical , meaning it is spread out evenly on both sides.

What is the difference between positive skew and negative skew?

With positive skew most of the values sit at the lower end and the tail stretches out to the right. With negative skew most of the values sit at the higher end and the tail stretches out to the left. In each case the long tail points in the direction the skew is named after.

True or False? In a perfectly symmetrical distribution the mean, the median and the mode are all equal.

True. A symmetrical distribution does not lean either way, so its centre comes out the same however you choose to measure it. That is why a gap between the mean and the median is itself a sign that the data is skewed.

How can you judge the skew of a distribution from a stem-and-leaf diagram?

Look at the length of each row of leaves , which works in the same way as the height of a bar on a chart. Rows that get steadily longer or shorter as you read down the diagram show the distribution leaning one way, with the tail at the end where the rows are shortest.

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Using Summary Statistics (AQA GCSE Statistics: Foundation): Flashcards

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FrontUsing Measures of Central Tendency
An ice cream seller records how many of each flavour were sold. Why is the mode the only average that can be used on this data?

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An ice cream seller records how many of each flavour were sold. Why is the mode the only average that can be used on this data?
Because the data values are the flavours, which are qualitative, and there is no way to add up or order a list of names.
The numbers sold are the frequencies rather than the data values, which is what makes this data easy to mistake for numerical data.
Ten employees earn between £256 and £458 except one who earns £3850. The median wage is £401 and the mean is £722.60. Which average describes the wages better?
The median, because nine of the ten employees earn far less than the mean does.
The single very large wage drags the mean upwards until it no longer describes a typical employee at all, while the median is left untouched by it.
Complete the comparison of the three averages.
The $______$ is the only average that uses every data value in its calculation, while the $______$ is the only one that is always a value from the data set itself.
The completed comparison is:
The mean is the only average that uses every data value in its calculation, while the mode is the only one that is always a value from the data set itself.
Why is the mean unreliable when a grouped data set has an open-ended class?
Because an open-ended class has no upper boundary, so there is no midpoint to put into the calculation.
Any value chosen for it is a guess, and since the mean depends on every class the whole estimate becomes unreliable.
True or False?
The median can be used to work out other statistics, such as the quartiles and the skew.
True.
The median is the starting point for the quartiles and therefore for the interquartile range, and it is used again when deciding how a distribution is skewed.
The mode, by contrast, cannot be used to work out an associated measure of spread at all.
When comparing two data sets, why must you use the same average for both?
Because a mean and a median measure different things, so setting one against the other says nothing about how the two data sets actually differ.
Only the same average, worked out the same way for both sets, gives a fair comparison.
True or False?
Two people comparing the same pair of data sets must reach the same conclusion.
False.
The conclusion depends on which average is chosen, and different averages can point in different directions when the data is skewed.
Someone with an argument to make, such as a politician, may pick whichever average supports it best.
What two things must you compare when comparing two data sets?
An average, which is a measure of central tendency, and a measure of spread.
One on its own is never enough, because two data sets can share the same average and still be spread out completely differently.
Complete the rule for pairing a measure of spread with an average.
If you compare the modes you should use the $______$ , and if you compare the medians you may use the range or the $______$ range.
The completed rule is:
If you compare the modes you should use the range, and if you compare the medians you may use the range or the interquartile range.
The interquartile range is the usual choice alongside a median.
Why should the interquartile range not be used alongside the mean?
Because the mean uses every value in the data set, while the interquartile range deliberately ignores the top and bottom quarters of it.
The two describe different parts of the data, so the range, which also depends on the extreme values, is what pairs with the mean.
What four things should a written comparison of two data sets contain?
A comparison of the two averages as numbers, followed by what that difference means in the context of the question.
Then a comparison of the two measures of spread as numbers, followed by what that difference means in context as well.
The numbers alone do not answer the question, so each one has to be turned back into a statement about the real situation.
Give three different ways of saying that one data set has a smaller spread than another.
Its values are less spread out, or they are closer together.
You can also say that the values are more consistent, or that there is less variation between them.
Why might a conclusion drawn from comparing two data sets still be unreliable?
The data sets may be too small to represent what they are meant to, since five pupils' heights say nothing about a whole school.
They may also be biased, for example if only the older year groups were measured, which would push the average height up.
Why might you compare data from a sample with data from the whole population?
To judge how representative the sample is.
If the sample's average and spread come out close to the population's, it is a sound basis for drawing conclusions, and if they are far apart it is not.
True or False?
A teacher and an examiner might want opposite things from the spread of a set of marks.
True.
A teacher may want a small spread, because it suggests the whole class has understood the topic to a similar standard.
An examiner may want a large spread, because it makes the grade boundaries easier to place.
Define skewness.
Skewness describes the way the data in a distribution leans, rather than where its centre lies.
A distribution can be skewed towards one side or the other, or it can be symmetrical, meaning it is spread out evenly on both sides.
What is the difference between positive skew and negative skew?
With positive skew most of the values sit at the lower end and the tail stretches out to the right.
With negative skew most of the values sit at the higher end and the tail stretches out to the left.
In each case the long tail points in the direction the skew is named after.
Complete the ordering of the three averages that indicates positive skew.
$mean______median______mode$
The completed ordering is:
$mean > median > mode$
Negative skew reverses it, giving mode greater than median greater than mean.
How do the median and the quartiles show which way a distribution is skewed?
Work out $median - LQ$ and compare it with $UQ - median$ to see which is larger.
If the first is the smaller of the two the median sits nearer the lower quartile and the skew is positive, and if it is the larger the skew is negative.
True or False?
In a perfectly symmetrical distribution the mean, the median and the mode are all equal.
True.
A symmetrical distribution does not lean either way, so its centre comes out the same however you choose to measure it.
That is why a gap between the mean and the median is itself a sign that the data is skewed.
How can you judge the skew of a distribution from a stem-and-leaf diagram?
Look at the length of each row of leaves, which works in the same way as the height of a bar on a chart.
Rows that get steadily longer or shorter as you read down the diagram show the distribution leaning one way, with the tail at the end where the rows are shortest.

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