What does the standard deviation tell you about a data set?

It measures spread , telling you how far the data values lie from the mean . A larger standard deviation means the data is more spread out, and a smaller one means the values sit closer to the mean.

True or False? A standard deviation of zero means the data set contains no values.

False. A standard deviation of zero means every data value is equal to the mean . The data set still has values in it, they simply do not vary at all.

How can you organise the working for a standard deviation?

Set out a table with one column for each quantity the formula needs, then total each column. Round the final answer sensibly, with three significant figures a good choice when the question does not say otherwise.

Define the median of a data set.

The median is an average , the middle value once the data has been put in numerical order. Putting the values in order first is essential, since the middle of an unordered list means nothing.

When splitting data into halves for the quartiles what happens to the median?

With an odd number of values the median is left out of both halves. With an even number there is no middle value to leave out, so every data value belongs to one half or the other.

Once the data is split into halves how do you find each quartile?

Find the median of each half , in exactly the way you found the median of the whole data set. The median of the lower half is the lower quartile, and the median of the upper half is the upper quartile.

Ten planks measure 90, 95, 100, 100, 105, 110, 110, 115, 120 and 125 cm, so what is the interquartile range?

The lower half is 90, 95, 100, 100, 105 giving a lower quartile of 100, and the upper half gives an upper quartile of 115. The interquartile range is therefore 115-100 = 15 cm.

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

Compare an average , and compare a spread , doing both rather than only one. The average is the mean or the median, and the spread is the standard deviation or the interquartile range.

True or False? The mean can be used together with the interquartile range when comparing data sets.

False. The two that belong together are the mean and standard deviation , or the median and interquartile range . Taking the average from one pair and the spread from the other is not accepted.

How many parts should a conclusion comparing two data sets have?

Two , one comparing the averages and one comparing the spreads. Each part has to be written in the context of the question rather than as a bare statement about numbers.

Why is it not enough to say that one data set has a higher mean than the other?

A comparison has to describe what the difference means in the real-life situation the question is about. Use the question's own wording, so a higher mean test score becomes "on average class A performed better than class B".

What does a smaller measure of spread tell you about a data set?

The values are less varied , so they sit closer together. Phrases such as "more consistent", "closer together" and "less spread out" all describe the same thing.

How do you know whether to use the mean or the median in a comparison?

The question tells you which average and which measure of spread to work out. You are never left to choose between them, so read the question rather than deciding for yourself.

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Data Sets (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontMean & Standard Deviation
Define the mean of a data set.

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Cards in this collection (23)

Define the mean of a data set.
The mean is an average, found by dividing the sum of the data values by the number of values.
It gives a typical value for the data set, so the mean of $1, 2, 6$ is $\frac{9}{3} = 3$ .
Complete the formula for the mean, which is not given to you in the exam.
$\bar{x} = \frac{Σ______}{______}$
The completed formula is:
$\bar{x} = \frac{Σ x}{n}$
Here $Σ x$ is the sum of all the data values and $n$ is how many values there are.
What does the standard deviation tell you about a data set?
It measures spread, telling you how far the data values lie from the mean.
A larger standard deviation means the data is more spread out, and a smaller one means the values sit closer to the mean.
True or False?
A standard deviation of zero means the data set contains no values.
False.
A standard deviation of zero means every data value is equal to the mean.
The data set still has values in it, they simply do not vary at all.
Both standard deviation formulae are given, so how do you choose?
Either gives the same answer, so choose on convenience.
If you have already worked out the mean, $s = \sqrt{\frac{Σ {(x - \bar{x})}^{2}}{n - 1}}$ is usually easier, and if you have not, the other form works from $Σ x$ and $Σ x^{2}$ alone.
What is the difference between $Σ x^{2}$ and ${(Σ x)}^{2}$ ?
In $Σ x^{2}$ you square first and add afterwards, squaring every value and totalling the squares.
In ${(Σ x)}^{2}$ you add first and square afterwards, so the single total is squared once.
In both standard deviation formulae what does $n$ stand for?
$n$ is the sample size, meaning the number of values in the data set.
What appears underneath is $n - 1$ rather than $n$ , so a sample of seven values divides by $6$ .
How can you organise the working for a standard deviation?
Set out a table with one column for each quantity the formula needs, then total each column.
Round the final answer sensibly, with three significant figures a good choice when the question does not say otherwise.
Define the median of a data set.
The median is an average, the middle value once the data has been put in numerical order.
Putting the values in order first is essential, since the middle of an unordered list means nothing.
How do you find the median when there is an even number of values?
Take the midpoint of the middle two values, by adding them together and dividing by $2$ .
For $1, 2, 3, 4$ the middle two values are $2$ and $3$ , so the median is $2.5$ .
Define the lower quartile and the upper quartile.
The lower quartile lies a quarter of the way along the ordered data, with $25 %$ of the values below it.
The upper quartile lies three quarters of the way along, with $75 %$ of the values below it.
When splitting data into halves for the quartiles what happens to the median?
With an odd number of values the median is left out of both halves.
With an even number there is no middle value to leave out, so every data value belongs to one half or the other.
Once the data is split into halves how do you find each quartile?
Find the median of each half, in exactly the way you found the median of the whole data set.
The median of the lower half is the lower quartile, and the median of the upper half is the upper quartile.
Complete the formula for the interquartile range.
$IQR =______quartile -______quartile$
The completed formula is:
$IQR = upper quartile - lower quartile$
Show this subtraction in your working rather than simply writing down the answer.
True or False?
The interquartile range is not affected by extreme values in a data set.
True.
The interquartile range measures only the middle $50 %$ of the data, so values at the very top or bottom fall outside it.
That is a difference from the standard deviation, which every value in the data set contributes to.
Ten planks measure 90, 95, 100, 100, 105, 110, 110, 115, 120 and 125 cm, so what is the interquartile range?
The lower half is 90, 95, 100, 100, 105 giving a lower quartile of 100, and the upper half gives an upper quartile of 115.
The interquartile range is therefore 115-100 = 15 cm.
What two things must you compare when comparing two data sets?
Compare an average, and compare a spread, doing both rather than only one.
The average is the mean or the median, and the spread is the standard deviation or the interquartile range.
True or False?
The mean can be used together with the interquartile range when comparing data sets.
False.
The two that belong together are the mean and standard deviation, or the median and interquartile range.
Taking the average from one pair and the spread from the other is not accepted.
How many parts should a conclusion comparing two data sets have?
Two, one comparing the averages and one comparing the spreads.
Each part has to be written in the context of the question rather than as a bare statement about numbers.
Why is it not enough to say that one data set has a higher mean than the other?
A comparison has to describe what the difference means in the real-life situation the question is about.
Use the question's own wording, so a higher mean test score becomes "on average class A performed better than class B".
What does a smaller measure of spread tell you about a data set?
The values are less varied, so they sit closer together.
Phrases such as "more consistent", "closer together" and "less spread out" all describe the same thing.
School A scored a mean of 20 with a standard deviation of 3.4, and School B a mean of 23 with a standard deviation of 1.5, so complete the two comparisons.
On average School A scored $______$ than School B.
School A scores were $______$ varied than School B.
The completed comparisons are:
On average School A scored lower than School B.
School A scores were more varied than School B.
How do you know whether to use the mean or the median in a comparison?
The question tells you which average and which measure of spread to work out.
You are never left to choose between them, so read the question rather than deciding for yourself.

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