Define measure of central tendency .

A measure of central tendency is an average that describes where the centre of the data is ; the mean , the median and the mode are all examples. They are all measures of location , the wider term for a statistic saying where data sits in the number system; quartiles and percentiles are measures of location too, but they are not averages. Because there are three of them, "the average" is not specific enough in statistics: always say which one you mean.

Which of the three averages is affected by extreme values, and which are not?

The mean is affected, because it is calculated from every value in the data set, so one very large or very small value pulls it towards itself. The median is not, because it is a position in the ordered data rather than a calculation; an extreme value at one end moves it by at most one place. The mode is not affected either, but it has its own problems: a data set can have more than one mode, which makes it bimodal , or no mode at all, and a mode can sit nowhere near the centre of the data.

True or False? A data set with a larger range must also have a larger interquartile range.

False. The two measure different things and can move independently. A single extreme value at one end makes the range enormous while leaving the interquartile range completely unchanged, because that value lies outside the middle 50% of the data. This is exactly why the interquartile range is the more useful measure of spread when a data set contains extreme values.

What is the 70th percentile, and what is an interpercentile range?

Percentiles divide the data into 100 parts , so the 70th percentile is the value seven tenths of the way through the data: 70% of the data lies below it and 30% above it. An interpercentile range is the difference between two given percentiles, so, for example , the 20th to 80th interpercentile range is the 80th percentile minus the 20th percentile. The units are the same as the units of the original data, and the interquartile range is itself the 25th to 75th interpercentile range.

What does an ungrouped frequency table keep, and why does that matter for the averages?

An ungrouped frequency table keeps every original data value : it only records how many times each value occurred, so nothing is thrown away. That means the mean, median and mode found from it are the actual values for the data set, not estimates. It is normally used for discrete numerical data, and is quicker to work with than a raw list because the values have already been collated.

A frequency table for ungrouped data has its highest frequency, 16, in the column headed 39. What is the mode?

The mode is 39 . The mode is the data value that occurs most often, and 16 is the number of times it occurred rather than the value itself. Reading the frequency as the mode is the commonest slip here: the frequency row tells you where to look, and the value row gives you the answer.

Why does a grouped frequency table give a modal class rather than a mode?

Because grouping has hidden the individual data values: the table records only how many values fell in each class, so there is no way to tell which single value occurred most often. The most that can be said is which class the data is concentrated in, and that class is called the modal class.

True or False? Two data sets with the same mean must have the same standard deviation.

False. The mean is a measure of location and the standard deviation is a measure of spread , and they carry completely independent information. So, for example , 49, 50, 51 and 10, 50, 90 both have a mean of 50, but the second is far more spread out and has a much larger standard deviation. This is why a description of a data set should always give one of each: an average on its own says nothing about how varied the data is.

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Statistical Measures (AQA AS Maths: Statistics): Flashcards

Exam code: 7356

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FrontBasic Statistical Measures
Define measure of central tendency.
BackBasic Statistical Measures
A measure of central tendency is an average that describes where the centre of the data is; the mean, the median and the mode are all examples.
They are all measures of location, the wider term for a statistic saying where data sits in the number system; quartiles and percentiles are measures of location too, but they are not averages.
Because there are three of them, "the average" is not specific enough in statistics: always say which one you mean.
BackBasic Statistical Measures
The mean is affected, because it is calculated from every value in the data set, so one very large or very small value pulls it towards itself.
The median is not, because it is a position in the ordered data rather than a calculation; an extreme value at one end moves it by at most one place.
The mode is not affected either, but it has its own problems: a data set can have more than one mode, which makes it bimodal, or no mode at all, and a mode can sit nowhere near the centre of the data.

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Cards in this collection (23)

Define measure of central tendency.
A measure of central tendency is an average that describes where the centre of the data is; the mean, the median and the mode are all examples.
They are all measures of location, the wider term for a statistic saying where data sits in the number system; quartiles and percentiles are measures of location too, but they are not averages.
Because there are three of them, "the average" is not specific enough in statistics: always say which one you mean.
Which of the three averages is affected by extreme values, and which are not?
The mean is affected, because it is calculated from every value in the data set, so one very large or very small value pulls it towards itself.
The median is not, because it is a position in the ordered data rather than a calculation; an extreme value at one end moves it by at most one place.
The mode is not affected either, but it has its own problems: a data set can have more than one mode, which makes it bimodal, or no mode at all, and a mode can sit nowhere near the centre of the data.
What do the symbols $Σ x$ and $\bar{x}$ stand for, and how is each read aloud?
$Σ x$ is the sum of all the data values, read "sigma $x$ ", and written out in full it is
$\sum_{i = 1}^{n} x_{i} = x_{1} + x_{2} + \dots + x_{n}$
$\bar{x}$ is the mean, read " $x$ bar", and the two are linked by
$\bar{x} = \frac{Σ x}{n}$
Statistics written this way are called summary statistics, because they summarise a whole data set in one number.
What do the three quartiles divide a data set into, and what does each one split?
The quartiles divide the data into four equal sections.
- The lower quartile $Q_{1}$ splits the lowest 25% from the highest 75%
- The median $Q_{2}$ is the value 50% of the way through the data
- The upper quartile $Q_{3}$ splits the lowest 75% from the highest 25%
Quartiles are measures of location, not averages: they say where a point in the data sits, not what is typical.
You have $n$ raw data values written in order. How do you find the position of the lower quartile?
Calculate $\frac{n}{4}$ , then look at what kind of number it is.
- If $\frac{n}{4}$ is not an integer, round it up to the next integer and take that value
- If $\frac{n}{4}$ is an integer, take the midpoint of that value and the one above it
So, for example, with $n = 7$ , $\frac{7}{4} = 1.75$ is not an integer, so $Q_{1}$ is the 2nd value; the upper quartile works the same way from $\frac{3 n}{4}$ .
This is the rule for raw values in order; estimating a quartile from grouped data uses a different rule, because the individual values are no longer known.
What is the difference between the range and the interquartile range?
Both are measures of spread, telling you how spread out the data is rather than where it is, and both carry the same units as the original data.
The range is the largest value minus the smallest value, so every data point contributes to it, including any extreme values.
The interquartile range is $Q_{3} - Q_{1}$ , so it covers only the middle 50% of the data and is not affected by extreme values.
True or False?
A data set with a larger range must also have a larger interquartile range.
False.
The two measure different things and can move independently.
A single extreme value at one end makes the range enormous while leaving the interquartile range completely unchanged, because that value lies outside the middle 50% of the data.
This is exactly why the interquartile range is the more useful measure of spread when a data set contains extreme values.
What is the 70th percentile, and what is an interpercentile range?
Percentiles divide the data into 100 parts, so the 70th percentile is the value seven tenths of the way through the data: 70% of the data lies below it and 30% above it.
An interpercentile range is the difference between two given percentiles, so, for example, the 20th to 80th interpercentile range is the 80th percentile minus the 20th percentile.
The units are the same as the units of the original data, and the interquartile range is itself the 25th to 75th interpercentile range.
What does an ungrouped frequency table keep, and why does that matter for the averages?
An ungrouped frequency table keeps every original data value: it only records how many times each value occurred, so nothing is thrown away.
That means the mean, median and mode found from it are the actual values for the data set, not estimates.
It is normally used for discrete numerical data, and is quicker to work with than a raw list because the values have already been collated.
Complete the formula for the mean of the data in a frequency table:
$\bar{x} = \frac{Σ______}{Σ______}$
The completed formula is:
$\bar{x} = \frac{Σ x f}{Σ f}$
Each value $x$ is multiplied by its frequency $f$ because that value occurs $f$ times, so $Σ x f$ is the total of all the data.
$Σ f$ is the total frequency, which is how many data values there are altogether, not how many different values appear in the table.
In an ungrouped frequency table of $n$ discrete values, how do you find which data value is the median?
Work out $\frac{n + 1}{2}$ to get the position of the median in the ordered data, then add the frequencies down the table until the running total first reaches that position.
The median is the data value in the row where that running total arrives, and if the position ends in $. 5$ you take the midpoint of the two values on either side.
Adding the frequencies is what carries you through the data in order: the table already lists the values in order, and the running total tells you how far through them you have got.
A frequency table for ungrouped data has its highest frequency, 16, in the column headed 39. What is the mode?
The mode is 39.
The mode is the data value that occurs most often, and 16 is the number of times it occurred rather than the value itself.
Reading the frequency as the mode is the commonest slip here: the frequency row tells you where to look, and the value row gives you the answer.
Why are classes for continuous data written as $10 \leq x < 20$ rather than as 10 to 20?
Because a table listing 10 to 20 and then 20 to 30 gives no way of deciding which class the value 20 belongs to, while the inequalities put every value in exactly one class.
If a grouped table does leave a gap between one class and the next, close it before doing any calculation: $10 \leq x \leq 19$ followed by $20 \leq x \leq 29$ becomes $9.5 \leq x < 19.5$ and $19.5 \leq x < 29.5$ .
The halfway points are used because the data has been rounded, so a value recorded as 19 could really be anything up to 19.5.
True or False?
In a table with classes $155 \leq h < 160$ and $160 \leq h < 165$ , a height of exactly 160 cm goes in the second class.
True.
The first class stops below 160, because $h < 160$ excludes 160 itself, while the second class starts at 160, because $160 \leq h$ includes it.
So the value belongs to $160 \leq h < 165$ , and reading the inequality signs rather than the numbers is what settles every case like this.
Why does a grouped frequency table give a modal class rather than a mode?
Because grouping has hidden the individual data values: the table records only how many values fell in each class, so there is no way to tell which single value occurred most often.
The most that can be said is which class the data is concentrated in, and that class is called the modal class.
How do you estimate the mean from a grouped frequency table, and what are you assuming when you do?
Use the midpoint of each class as its $x$ value, then work out the mean exactly as you would for an ungrouped table.
The assumption is that the values in each class average out at its midpoint, which they need not, so the result is an estimate rather than the true mean.
So, for example, the five midpoints in a height table total 4052.5 once each is multiplied by its frequency, and dividing by the total frequency of 25 gives an estimated mean of 162 cm.
Define variance.
The variance is a measure of spread: it measures how varied a set of data is about its mean.
Data that is spread out has a greater variance, and data whose values are close together has a smaller variance.
The symbol for the variance of a population is $σ^{2}$ , using the lower-case Greek letter sigma.
How are the standard deviation and the variance related, and why is the standard deviation usually the one quoted?
The standard deviation is the square root of the variance, so $σ = \sqrt{σ^{2}}$ , and the reason for preferring it is units.
Squaring the deviations to find the variance squares the units as well, so a variance of times in minutes comes out in minutes squared, which cannot be compared with the data.
Taking the square root puts them back, leaving the standard deviation in the same units as the original data; the two are used interchangeably in this course, so check which one a question gives you and which it wants.
Complete the version of the variance formula that is quickest to use in most questions:
$σ^{2} = \frac{Σ x^{2}}{n} - {(______)}^{2}$
The completed formula is:
$σ^{2} = \frac{Σ x^{2}}{n} - {(\bar{x})}^{2}$
An easy way to remember it is "the mean of the squares minus the square of the mean".
The order matters: subtracting the other way round would give a negative answer, and a variance can never be negative.
$σ^{2} = \frac{Σ {(x - \bar{x})}^{2}}{n}$ is the definition of the variance, so why is it rarely the version you use?
It is slow: you have to find the mean first, then go back through the data working out a separate squared deviation for every single value, and only then add them up.
The equivalent version $\frac{Σ x^{2}}{n} - {(\bar{x})}^{2}$ needs only two totals, $Σ x$ and $Σ x^{2}$ , and those are usually exactly what a question hands you as summary statistics, or what a calculator produces from the data.
The two forms always give the same answer: one is a rearrangement of the other.
How does the variance formula change when the data is given in a frequency table?
Every value is weighted by its frequency, and the total frequency $Σ f$ replaces $n$ :
$σ^{2} = \frac{Σ f x^{2}}{Σ f} - {(\frac{Σ f x}{Σ f})}^{2}$
It is the same formula, with the second bracket just $\bar{x}$ written out from the table; for a grouped table, $x$ is the midpoint of each class, so the answer is an estimate.
Be consistent about which statistics you use: if the $Σ f x^{2}$ you put in is calculated from coded or midpoint values, the mean you square must come from the same values.
What is the summary statistic $S_{x x}$ , and how does it give the variance?
$S_{x x}$ is the total squared deviation from the mean:
$S_{x x} = Σ {(x - \bar{x})}^{2} = Σ x^{2} - \frac{{(Σ x)}^{2}}{n}$
The variance is then $σ^{2} = \frac{S_{x x}}{n}$ , and the standard deviation is $\sqrt{\frac{S_{x x}}{n}}$ .
It is worth recognising, because it is the form the formula booklet uses and it turns up again in the formula for the product moment correlation coefficient.
True or False?
Two data sets with the same mean must have the same standard deviation.
False.
The mean is a measure of location and the standard deviation is a measure of spread, and they carry completely independent information.
So, for example, 49, 50, 51 and 10, 50, 90 both have a mean of 50, but the second is far more spread out and has a much larger standard deviation.
This is why a description of a data set should always give one of each: an average on its own says nothing about how varied the data is.

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