What is the mode, and what do you say if no value repeats?

The mode is the value that appears most often, and there can be more than one. If no value repeats there is simply no mode , and you must not write that the mode is zero, since zero would be a value like any other.

How do you find the median of an even number of values?

Put the values in size order and take the midpoint of the middle two. That means adding those two values and dividing by 2, so the median of 1, 2, 3, 4 is 2.5.

True or False? A mean must be rounded to a whole number when the data is a count of people.

False. A mean can be a fraction or a decimal, and a mean of 7.5 people is a perfectly correct answer. Round only when the question tells you to, and to the accuracy it asks for.

When should you use the median rather than the mean?

When the data contains extreme values . A single unusually large or small value drags the mean towards it, while the median stays where it is: in 1, 1, 2, 3, 5, 5, 5, 25 the 25 distorts the mean but leaves the median untouched.

When is the mode the only average you can use?

When the data is non-numerical , such as types of pet or favourite colours. You cannot add such data to get a mean, and you cannot put it in size order to get a median, but you can still count which category occurs most often.

The quartiles are the values that split an ordered data set into four equal parts. The lower quartile lies a quarter of the way along, so 25% of the data is below it, and the upper quartile lies three quarters of the way along, so 75% is below it.

When splitting an odd number of values into two halves for the quartiles, what happens to the median?

The median is not included in either half. The lower half is everything below it and the upper half is everything above it, so one value is set aside. With an even number of values there is no such value to set aside, and every value belongs to one half or the other.

Once the data is split into two halves, how do you find the quartiles?

The lower quartile is the median of the lower half. The upper quartile is the median of the upper half. So the same middle-value procedure is carried out three times in total: once on the whole set and once on each half.

What is the interquartile range, and what does it measure?

The interquartile range is the upper quartile minus the lower quartile. It measures how spread out the middle 50% of the data is, ignoring the quarter at each end.

True or False? The interquartile range is affected by extreme values in the data.

False. It uses only the middle 50%, so a single very large or very small value sits outside it and changes nothing. The range is affected, because it is worked out from the highest and lowest values themselves, and this is the reason both measures exist.

What does the standard deviation tell you?

How spread out the data is around the mean . A larger standard deviation means the values are more spread out, and a smaller one means they are bunched closer together.

True or False? The two given standard deviation formulae always give the same answer.

True. They are two ways of writing the same calculation, so neither is more accurate than the other. Which one to use is purely a question of which is quicker with the numbers you have been given.

What two things must a comparison of two data sets include?

A comparison of their averages and a comparison of their spreads . One without the other is not a full comparison: two data sets can share an average and still be quite different in how varied they are.

True or False? The data set with the higher average is automatically the better one.

False. Whether higher is better depends entirely on what is being measured. For times taken to complete a puzzle a lower average is the better result, so the context has to be read before any judgement is made.

What makes a comparison of two averages a valid one?

It has to say what is higher or lower, in context, and use the word average . "On average, students in class A scored higher on the test than students in class B" does that; "on average, class A is higher" does not, because it never says the scores are what is higher.

How should a comparison of two spreads be worded?

In terms of how varied or consistent the data is, again in context. "The scores in class A were more consistent" is a comparison of the data, whereas "the standard deviation for class A is lower" only compares the two statistics and leaves the reader to work out what that means.

School A had a mean of 20 correct answers with standard deviation 3.4, and School B a mean of 23 with standard deviation 1.5. What two comments can you make?

On average, the number of correct answers was lower in School A than in School B. The numbers of correct answers were more varied in School A than in School B. The mean is paired with the standard deviation throughout, and each comment names what is being compared.

True or False? A smaller spread means the data is more consistent.

True. A small standard deviation or interquartile range means the values sit close to one another. That is exactly what the word consistent describes, which is why it is the right word to reach for when the spread is the smaller of the two.

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Comparing Data (SQA National 5 Applications of Mathematics): Flashcards

Exam code: X844 75

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What is the mode, and what do you say if no value repeats?

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What is the mode, and what do you say if no value repeats?
The mode is the value that appears most often, and there can be more than one.
If no value repeats there is simply no mode, and you must not write that the mode is zero, since zero would be a value like any other.
How do you find the median of an even number of values?
Put the values in size order and take the midpoint of the middle two.
That means adding those two values and dividing by 2, so the median of 1, 2, 3, 4 is 2.5.
Complete the formula for the mean, which is not given to you in the exam:
$\bar{x} = \frac{Σ______}{______}$
The completed formula is:
$\bar{x} = \frac{Σ x}{n}$
$Σ x$ is the sum of all the data values and $n$ is how many values there are.
True or False?
A mean must be rounded to a whole number when the data is a count of people.
False.
A mean can be a fraction or a decimal, and a mean of 7.5 people is a perfectly correct answer.
Round only when the question tells you to, and to the accuracy it asks for.
When should you use the median rather than the mean?
When the data contains extreme values.
A single unusually large or small value drags the mean towards it, while the median stays where it is: in 1, 1, 2, 3, 5, 5, 5, 25 the 25 distorts the mean but leaves the median untouched.
When is the mode the only average you can use?
When the data is non-numerical, such as types of pet or favourite colours.
You cannot add such data to get a mean, and you cannot put it in size order to get a median, but you can still count which category occurs most often.
Fifteen times were recorded and add up to 226 seconds. What is the mean, to 3 significant figures?
The mean time is 15.1 seconds.
Dividing the total by the number of values gives $226 \div 15 = 15.0666 . . .$
The units belong in the answer: a mean of a set of times is itself a time.
Complete the definition of the range:
The range is the $______$ value minus the $______$ value, and it measures how spread out the data is.
The completed definition is:
The range is the highest value minus the lowest value, and it measures how spread out the data is.
Comparing the ranges of two data sets shows which of them is more spread out.
What is the range of $- 2$ , $- 1$ , 0 and 4?
The range is 6.
Subtracting the lowest from the highest gives $4 - (- 2) = 6$ , because subtracting a negative adds.
Writing 4 minus 2 here is the usual slip, and it gives 2 rather than 6.
Define quartiles.
The quartiles are the values that split an ordered data set into four equal parts.
The lower quartile lies a quarter of the way along, so 25% of the data is below it, and the upper quartile lies three quarters of the way along, so 75% is below it.
When splitting an odd number of values into two halves for the quartiles, what happens to the median?
The median is not included in either half.
The lower half is everything below it and the upper half is everything above it, so one value is set aside.
With an even number of values there is no such value to set aside, and every value belongs to one half or the other.
Once the data is split into two halves, how do you find the quartiles?
The lower quartile is the median of the lower half.
The upper quartile is the median of the upper half.
So the same middle-value procedure is carried out three times in total: once on the whole set and once on each half.
What is the interquartile range, and what does it measure?
The interquartile range is the upper quartile minus the lower quartile.
It measures how spread out the middle 50% of the data is, ignoring the quarter at each end.
True or False?
The interquartile range is affected by extreme values in the data.
False.
It uses only the middle 50%, so a single very large or very small value sits outside it and changes nothing.
The range is affected, because it is worked out from the highest and lowest values themselves, and this is the reason both measures exist.
The lower half of an ordered data set is 31, 32, 35, 35, 36, 37, 39, 40, 42, 45. What is the lower quartile?
The lower quartile is 36.5.
It is the median of these ten values, so it is the midpoint of the fifth and sixth: 36 and 37.
Taking their midpoint gives $(36 + 37) \div 2 = 36.5$ .
What does the standard deviation tell you?
How spread out the data is around the mean.
A larger standard deviation means the values are more spread out, and a smaller one means they are bunched closer together.
The formulae list gives two formulae for the standard deviation. What does each one need?
The first needs the mean worked out first, and then the squared difference between the mean and every value.
The second needs only $Σ x$ and $Σ x^{2}$ , so it avoids finding the mean at all and is often quicker to tabulate.
Complete the divisor used in both standard deviation formulae:
In both formulae you divide by $______$ rather than by $n$ , because it is one $______$ than the number of values.
The completed sentence is:
In both formulae you divide by n − 1 rather than by $n$ , because it is one less than the number of values.
With seven values you divide by 6, and using 7 by mistake makes the answer slightly too small.
Using the formula with the mean, what do you do once the mean is found?
Subtract the mean from each value to get the differences.
Square every one of those differences, which also makes the negative ones positive.
Add the squares together, and that total is $Σ {(x - \bar{x})}^{2}$ .
Seven values have a mean of 20 and give $Σ {(x - \bar{x})}^{2} = 68$ . What is the standard deviation?
The standard deviation is 3.4.
Dividing by one less than the number of values gives $68 \div 6 = 11.33 . . .$
Taking the square root gives $\sqrt{11.33 . . .} = 3.3665 . . .$ , which rounds to 3.4.
True or False?
The two given standard deviation formulae always give the same answer.
True.
They are two ways of writing the same calculation, so neither is more accurate than the other.
Which one to use is purely a question of which is quicker with the numbers you have been given.
If you have already worked out the mean, how can you get $Σ x$ quickly?
Multiply the mean by the number of values.
The mean is $Σ x$ divided by $n$ , so multiplying it back by $n$ returns the total, which saves adding the values a second time.
What two things must a comparison of two data sets include?
A comparison of their averages and a comparison of their spreads.
One without the other is not a full comparison: two data sets can share an average and still be quite different in how varied they are.
Complete the pairing rule for comparing data sets:
You compare either the mean alongside the $______$ , or the median alongside the $______$ , and the two pairs are never mixed.
The completed rule is:
You compare either the mean alongside the standard deviation, or the median alongside the interquartile range, and the two pairs are never mixed.
Each pair belongs together: the mean and standard deviation both work from distances to the mean, while the median and interquartile range are both about position in an ordered list.
True or False?
The data set with the higher average is automatically the better one.
False.
Whether higher is better depends entirely on what is being measured.
For times taken to complete a puzzle a lower average is the better result, so the context has to be read before any judgement is made.
What makes a comparison of two averages a valid one?
It has to say what is higher or lower, in context, and use the word average.
"On average, students in class A scored higher on the test than students in class B" does that; "on average, class A is higher" does not, because it never says the scores are what is higher.
How should a comparison of two spreads be worded?
In terms of how varied or consistent the data is, again in context.
"The scores in class A were more consistent" is a comparison of the data, whereas "the standard deviation for class A is lower" only compares the two statistics and leaves the reader to work out what that means.
School A had a mean of 20 correct answers with standard deviation 3.4, and School B a mean of 23 with standard deviation 1.5. What two comments can you make?
On average, the number of correct answers was lower in School A than in School B.
The numbers of correct answers were more varied in School A than in School B.
The mean is paired with the standard deviation throughout, and each comment names what is being compared.
True or False?
A smaller spread means the data is more consistent.
True.
A small standard deviation or interquartile range means the values sit close to one another.
That is exactly what the word consistent describes, which is why it is the right word to reach for when the spread is the smaller of the two.

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