Statistics 1 Chapter 2 ::� Measures of Location & Spread

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Experimental

i.e. Dealing with collected data.

Theoretical

Deal with probabilities and modelling to make inferences about what we ‘expect’ to see or make predictions, often using this to reason about/contrast with experimentally collected data.

Chp1: Data Collection

Methods of sampling, types of data, and populations vs samples.

Chp2: Measures of Location/Spread

Statistics used to summarise data, including mean, standard deviation, quartiles, percentiles. Use of linear interpolation for estimating medians/quartiles.

Chp3: Representation of Data

Producing and interpreting visual representations of data, including box plots and histograms.

Chp5: Probability

Venn Diagrams, mutually exclusive + independent events, tree diagrams.

Chp6: Statistical Distributions

Common distributions used to easily find probabilities under certain modelling conditions, e.g. binomial distribution.

Chp7: Hypothesis Testing

Determining how likely observed data would have happened ‘by chance’, and making subsequent deductions.

Chp4: Correlation

Measuring how related two variables are, and using linear regression to predict values.

This is identical to the equivalent chapter in the old S1 module. Some content will be familiar from GCSE (mean of grouped/ungrouped data), but many concepts new (e.g. standard deviation) along with possibly unfamiliar notation.

“Calculate the mean of this grouped frequency table.”

1:: Mean, Median, Mode

“Use linear interpolation to estimate the interquartile range.”

2:: Quartiles, Percentiles, Deciles

“Calculate the standard deviation of the maths marks.”

3:: Variance & Standard Deviation

4:: Coding

Similarities

Differences

Measures of Central Tendency

Measures of Location

Measures of Spread

Range

Interquartile Range

Standard Deviation

Variance

Mode

Median

Mean

Quartiles

Percentiles

Deciles

Maximum

Minimum

Measures of location are single values which describe a position in a data set.

Of these, measures of central tendency are to do with the centre of the data, i.e. a notion of ‘average’.

Measures of spread are to do with how data is spread out.

You all know how to find the mean of a list of values. But lets consider the notation, and see how theoretically we could calculate each of the individual components on a calculator.

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The ‘overbar’ in stats specifically means ‘the sample mean of’, but don’t worry about the ‘sample’ bit for now.

“Use of Technology” Monkey says:

Time to whip out yer Casios…

Use the MENU button to access STATS mode.

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How on the calculator would we get…

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Why is our mean just an estimate?

Because we don’t know the exact heights within each group. Grouping data loses information.

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Use your calculator’s STATS mode to determine the mean (or estimate of the mean).

Ensure that you show the division in your working.

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Pearson Statistics 1

Pages 8, 10-11

The mean maths score of 20 pupils in class A is 62.

The mean maths score of 30 pupils in class B is 75.

1. What is the overall mean of all the pupils’ marks.
2. The teacher realises they mismarked one student’s paper; he should have received 100 instead of 95. Explain the effect on the mean and median.

Archie the Archer competes in a competition with 50 rounds. He scored an average of 35 points in the first 10 rounds and an average of 25 in the remaining rounds. What was his average score per round?

Test Your Understanding

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This subtopic doesn’t appear in your textbook but has cropped up in exams.

You need to be able to find the median of both listed data and of grouped data.

Listed data

Grouped data

Position to use for median:

8.5

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What position do we use for the median?

Median position: 6^th

Median position: 12^th/13^th

Median position: 8.5

Median position: 5

Median position: 30^th/31^st

Median position: 10.5

Median position: 18^th

Median position: 6.5

Median position: 9^th/10^th

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At GCSE we could find the median by drawing a suitable line on a cumulative frequency graph. How could we read off this value exactly using a suitable calculation?

We could find the fraction of the way along the line segment using the frequencies, then go this same fraction along the class interval.

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55

100

75

0.6m

Med

0.65m

Frequency up until this interval

Frequency by end of this interval

Item number we’re interested in.

Height at start of interval.

Height by end of interval.

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55

100

75

0.6m

Med

0.65m

Frequency up until this interval

Frequency by end of this interval

Item number we’re interested in.

Height at start of interval.

Height by end of interval.

Fro Tip: I like to put the units to avoid getting frequencies confused with values of the variable.

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Fro Tip: To quickly get frequency before and after, just look for the two cumulative frequencies that surround the item number.

10

18

16

3kg

4kg

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20

10

12s

14s

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Med

Med

There are GAPS between intervals!

What interval does this actually represent?

Lower class boundary

Upper class boundary

Class width = 3

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Class width = 10

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Lower class boundary = 200

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Class width = 3

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Lower class boundary = 3.5

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Class width = 2

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Lower class boundary = 29

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Class width = 10

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Lower class boundary = 30.5

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Edexcel S1 Jan 2007 Q4

10

29

72

97

105

111

116

119

120

29

72

60

19.5 miles

29.5 miles

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Med

Questions should be on a printed sheet…

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Pearson Statistics 1

Pages 12-13

Q1a, 3, 4a, 5a

There is also a supplementary worksheet consisting of S1 exam questions (see next slides).

Questions should be on a printed sheet…

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You need to be able to find the quartiles of both listed data and of grouped data. The rule is exactly the same as for the median.

Listed data

Grouped data

Position to use for LQ:

4.25

Again, DO NOT round this value.

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The LQ, median and UQ give you 25%, 50% and 75% along the data respectively.

But we can have any percentage you like!

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You will always find these for grouped data in an exam, so never round this position.

Lower Quartile:

Median:

Upper Quartile:

57^th Percentile:

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Notation:

The interquartile range and interpercentile range are examples of measures of spread.

Bottom 25% of data

Middle 50% of data

Top 25% of data

Interquartile Range

Range

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Min

Max

These are the same as the ‘Test Your Understanding’ questions on your supplementary sheet from before.

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Pearson Statistics 1

Pages 12-13

Pages 15-16

Ex 2C: Q1b, 2, 4b-d, 5b-c, 6

Ex 2D

Again, there is also a supplementary worksheet consisting of S1 exam questions (see next slides).

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Distribution of IQs in L6Ms4

Distribution of IQs in L6Ms5

Here are the distribution of IQs in two classes of the same size. What’s the same, and what’s different?

The (estimated) mean IQ is the same for the two classes, as is the (estimated) range, but the overall spread of values is greater for the second class.

The interquartile range would convey this, but do we have a method of measuring spread that takes into account all the values?

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Variance is a measure of spread that takes all values into account.

Variance, by definition, is the average squared distance from the mean.

Distance from mean…

Squared distance from mean…

Average squared distance from mean…

“The mean of the squares minus the square of the mean (‘msmsm’)”

Variance

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Standard Deviation

The standard deviation can ‘roughly’ be thought of as the average distance from the mean.

But in practice you will never use this form, and it’s possible to simplify the formula to the following*:

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2cm 3cm 3cm 5cm 7cm

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3, 11

So note that that in the case of two items, the standard deviation is indeed the average distance of the values from the mean.

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Find the variance and standard deviation of the following sets of data.

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We can just mull over our mnemonic again:

Variance: “The mean of the squares minus the square of the mean (‘msmsm’)”

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Fro Tip: It’s better to try and memorise the mnemonic than the formula itself – you’ll understand what’s going on better.

Fro Exam Note: In an exam, you will pretty much certainly be asked to find the standard deviation for grouped data, and not listed data.

May 2013 Q4

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May 2013 (R) Q3

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Pearson Statistics 1S

Pages 17

What do you reckon is the mean height of people in this room?

Now, stand on your chair, as per the instructions below.

INSTRUCTIONAL VIDEO

Is there an easy way to recalculate the mean based on your new heights? And the variance of your heights?

The mean would increase by the height of the chairs. The spread however is unaffected thus the variance would remain the same.

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Suppose now after a bout of ‘stretching you to your limits’, you’re now all 3 times your original height.

What do you think happens to the standard deviation of your heights?

It becomes 3 times larger (i.e. your heights are 3 times as spread out!)

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What do you think happens to the variance of your heights?

It becomes 9 times larger. We use the scale factor of the standard deviation, squared.

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Coding

As discussed, adding (and subtracting) has no effect on standard deviation or any measure of spread.

Standard deviation will get 3 times larger.

-5 has no effect but standard deviation will get 2 times larger.

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£1010 £1020 £1030 £1040 £1050

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We ‘code’ our variable using the following:

£1 £2 £3 £4 £5

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The jist of coding: We want to find the mean/standard deviation of a variable. We transform the values, using some rule, to make them simpler. We can then more easily calculate the mean/standard deviation of the ‘coded’ data, and from this we can then determine what the mean/standard deviation would have been for the original uncoded data.

Coding

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Suppose we’ve worked all these out already.

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Pearson Statistics 1

Pages 20

For the following grouped frequency table, calculate:

a) The estimate mean:

b) The estimate median:

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