S1: Chapter 2/3 Data: Measures of Location and Dispersion Dr J Frost Last modified: 9 th September 2015

For Teacher Use

Variables in algebra vs stats Similarities Differences Discrete ? Continuous ?

Mean of ungrouped data 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. ? ?  Whip out yer Casios… The overbar in stats specifically means ‘the sample mean of’, but don’t worry about this for now.

Inputting Data 1.Press the ‘MODE’ button and select STAT. 2.The “1-VAR” means “one variable”, so select this. (We’ll also be using A + BX later in the year, which as you might guess, is to do with fitting straight lines of best fit) 3.Enter each value above in the table that appears, and press ‘=‘ after each one. 4.Now press AC to stop entering data so we can start inputting a calculation…

Inputting Data ? ? ? ? ?

Grouped Data Frequency Why is our mean just an estimate? The midpoints of each interval. They‘re effectively a sensible single value used to represent each interval. Because we don’t know the exact heights within each group. Grouping data loses information. ??? ? ?

Inputting Frequency Tables Frequency We now need to input frequencies. Go to SETUP (SHIFT -> MODE), press down, then select STAT. Select ‘ON’ for frequency. Your calculator will preserve this setting even when switched off. ? ?

Mini-Exercise Use your calculator’s STAT mode to determine the mean (or estimate of the mean). Ensure that you show the division in your working. ? ? ? 1 2 3

GCSE RECAP :: Combined Mean The mean maths score of 20 pupils in class A is 62. The mean maths score of 30 pupils in class B is 75. a)What is the overall mean of all the pupils’ marks. b)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 ? ?

Median – which item? You need to be able to find the median of both listed data and of grouped data. Listed data ItemsPosition of medianMedian 53 rd 7 42 nd /3 rd th th /6 th 7.5 Grouped data Position to use for median: 8.5 ?? ?? ?? ?? ? ?

Quickfire Questions… What position do we use for the median? Median position: 6 th Median position: 12 th /13 th AgeFreq Median position: 8.5 ScoreFreq Median position: 5 Bro Exam Note: Pretty much every exam question has dealt with median for grouped data, not listed data. Median position: 30 th /31 st ScoreFreq Median position: 10.5 Median position: 18 th Volume (ml)Freq Median position: 6.5 Median position: 9 th /10 th ? ? ? ? ? ? ? ? ?

Linear Interpolation Height of tree (m)FreqC.F. At GCSE we draw a suitable line on a cumulative frequency graph. How could we read of this value exactly using 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.

Linear Interpolation Height of tree (m)FreqC.F mMed0.65m Frequency up until this interval Frequency at end of this interval Item number we’re interested in. Height at start of interval.Height at end of interval. ? ? ? ? ?

Linear Interpolation mMed0.65m Frequency up until this interval Frequency at end of this interval Item number we’re interested in. Height at start of interval.Height at end of interval. Height of tree (m)FreqC.F. Bro Tip: I like to put the units to avoid getting frequencies confused with values. ? ? Bro Tip: To quickly get frequency before and after, just look for the two cumulative frequencies that surround the item number.

Weight of cat (kg)FreqC.F. More Examples kg4kg ? ??? ?? ? ? Time (s)FreqC.F s14s ? ? ? ?? ?

Weight of cat to nearest kgFrequency What’s different about the intervals here? There are GAPS between intervals! What interval does this actually represent? Lower class boundaryUpper class boundary Class width = 3 ? ?

Identify the class width … Class width = 10 ? … … … Lower class boundary = 200 ? Class width = 3 ? Lower class boundary = 3.5 ? Class width = 2 ? Lower class boundary = 29 ? Class width = 10 ? Lower class boundary = 30.5 ?

Linear Interpolation with gaps Jan 2007 Q miles 29.5 miles ?? ? ? ? ?

Test Your Understanding Age of relic (years)Frequency Questions should be on a printed sheet… Shark length (cm)Frequency ? ?

Exercise 2 Questions should be on a printed sheet… ? ? ?

Exercise 2 4 ? ?

Quartiles – which item? 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 ItemsPosition of LQ & UQLQ & UQ 52 nd & 4 th 7 41 st /2 nd & 3 rd /4 th 6.5 & nd & 6 th 4 & rd and 8 th 3 & 10 Grouped data Position to use for LQ: 4.25 Again, DO NOT round this value. ? ? ? ? ? ? ? ? ? ?

Percentiles The LQ, median and UQ give you 25%, 50% and 75% along the data respectively. But we can have any percentage you like ? You will always find these for grouped data in an exam, so never round this position. Lower Quartile: Median: Upper Quartile:57 th Percentile: ? ? ? ? Notation:

Test Your Understanding Age of relic (years)Frequency These are the same as the ‘Test Your Understanding’ questions on your sheet from before. Shark length (cm)Frequency ? ? ? ? ? ? ?

Exercise 3 ? ?

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What is variance? Distribution of IQs in L6Ms4 Distribution of IQs in L6Ms5 Here are the distribution of IQs in two classes. What’s the same, and what’s different?

Variance Variance is how spread out data is. Variance, by definition, is the average squared distance from the mean. Distance from mean… Squared distance from mean… Average squared distance from mean…

Simpler formula for variance “The mean of the squares minus the square of the mean (‘msmsm’)” Variance ? ? Standard Deviation The standard deviation can ‘roughly’ be thought of as the average distance from the mean.

Examples 2cm3cm3cm5cm7cm ? ? 1, 3 So note that that in the case of two items, the standard deviation is indeed the average distance of the values from the mean. ? ?

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

Extending to frequency/grouped frequency tables We can just mull over our mnemonic again: Variance: “The mean of the squares minus the square of the means (‘msmsm’)” ? ? Bro Tip: It’s better to try and memorise the mnemonic than the formula itself – you’ll understand what’s going on better, and the mnemonic will be applicable when we come onto random variables in Chapter 8. Bro Exam Note: In an exam, you will pretty much certainly be asked to find the standard deviation for grouped data, and not listed data.

Example May 2013 Q4 ? ? ?

Test Your Understanding May 2013 (R) Q3 ?

Most common exam errors

Exercises Page 40 Exercise 3C Q1, 2, 4, 6 Page 44 Exercise 3D Q1, 4, 5

Coding 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?

Starter 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!) ? What do you think happens to the variance of your heights? It becomes 9 times larger ? (Can you prove the latter using the formula for variance?)

The point of coding £1010 £1020 £1030 £1040 £1050 ? ? We ‘code’ our variable using the following: £1 £2 £3 £4 £5 ?

Finding the new mean/variance Coding ?? ?? ?? ?? ?? ??

Example Exam Question Suppose we’ve worked all these out already. ?

Exercises ? ? ? ?

Chapters 2-3 Summary For the following grouped frequency table, calculate: Frequency a) The estimate mean: b) The estimate median: ? ? ? ? ? ?

Chapters 2-3 Summary ? ? ? ? ?