GCSE: Constructions & Loci Dr J Frost ([email protected]) Last modified: 28 th December 2014.
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Transcript of S1: Chapter 2/3 Data: Measures of Location and Dispersion Dr J Frost ([email protected])...
![Page 1: S1: Chapter 2/3 Data: Measures of Location and Dispersion Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 9 th September.](https://reader036.fdocuments.us/reader036/viewer/2022062517/56649f135503460f94c27cc5/html5/thumbnails/1.jpg)
S1: Chapter 2/3Data: Measures of Location and Dispersion
Dr J Frost ([email protected])www.drfrostmaths.com
Last modified: 9th September 2015
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For Teacher UseLesson 1:
Concept of variables in statistics.Mean of ungrouped and grouped data.Using a calculator to input data and calculate .Combined means.
Lesson 2:Median from discrete data and continuous data (usinglinear interpolation)
Lesson 3:Quartiles/percentiles using linear interpolation.
Lesson 4:Variance/Standard Deviation
Lesson 5:Coding. Consolidation.
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Variables in algebra vs stats
Similarities
π₯ Just like in algebra, variables in stats
represent the value of some quantity, e.g. shoe size, height, colour.
Variables can be discrete or continuous.
Discrete: Limited to specific values, e.g. shoe size, colour, day born. May be quantitative (i.e. numerical) or qualitative (e.g. favourite colour).Continuous: Can be any real number in some range (time, weight, β¦)
Can be part of further calculations, e.g. if represents height, then represents twice peopleβs height. In stats this is known as βcodingβ, which weβll cover later.
Differences
Unlike algebra, a variable in stats represents the value of multiple objects. e.g. The heights of all people in a room.
Because of this, we can do operations on it as if it was a collection of values:
If represents peopleβs heights, gives the sum of everyoneβs heights.In algebra this would be meaningless. If , then makes no sense!
is the mean of . Notice is a collection of values whereas is a single value.
Discrete ?
Continuous ?
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Mean of ungrouped data
π₯=Ξ£π₯π
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.
Length of cat
?
?
Whip out yer Casiosβ¦
The overbar in stats specifically means βthe sample mean ofβ, but donβt worry about this for now.
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Length of cat
Inputting Data
1. Press the βMODEβ button and select STAT.2. The β1-VARβ means βone variableβ, so select this.
(Weβll 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 β=β after each one.
4. Now press AC to stop entering data so we can start inputting a calculationβ¦
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Length of cat
Inputting Data
In STATS mode, we can still do many calculations as normal (e.g. try 1 + 1 = ) but we can also insert various statistical calculations involving . Use SHIFT -> [1] to access these. Try inputting these expressions:
π=π Ξ£π₯=ππ .πππ₯=π .ππΞ£π₯ /π=π .ππ
3 π₯+1=π .ππ
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Grouped Data
Height of bear (in metres) Frequency
Estimate of Mean:
What does the variable represent?
Why is our mean just an estimate?
The midpoints of each interval. Theyβre effectively a sensible single value used to represent each interval.
Because we donβt know the exact heights within each group. Grouping data loses information.
? ? ?
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Inputting Frequency TablesHeight of bear (in metres) Frequency
We now need to input frequencies.Go to SETUP (SHIFT -> MODE), press down, then select STAT.Select βONβ for frequency.Your calculator will preserve this setting even when switched off.
Now go into STATS mode and input your table, using your midpoints as the values. How on the calculator do we get:
π₯=β ππ₯
β π
? This is . This is because your calculator thinks is the original variable rather than just a list of interval midpoints. Just imagine that your calculator is converting your table into a list of values with duplicates.
? This is . (Since )
Bro Exam Tip: In the exam you get a method mark for the division and an accuracy mark for the final answer. Write:
Youβre not required to show working like ββ
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Mini-Exercise
Num children () Frequency ()
Use your calculatorβs STAT mode to determine the mean (or estimate of the mean). Ensure that you show the division in your working.
π=Ξ£ππ
=1110
=1.1
IQ of L6Ms2 () Frequency ()
π₯=Ξ£ ππ₯Ξ£ π
=156016
=97.5
Time Frequency ()
to 3sf.
? ?
?
1 2
3
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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 pupilsβ marks. b) The teacher realises they mismarked one studentβs paper; he should have
received 100 instead of 95. Explain the effect on the mean and median.
The revised score will increase the mean but leave the median unaffected.
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
π΄πππ=(ππΓππ)+ (ππΓππ)
ππ=ππ
?
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Median β which item?You need to be able to find the median of both listed data and of grouped data.
Listed data
Items Position of median Median
5 3rd 7
4 2nd/3rd 9.5
7 4th 7
10 5th/6th 7.5
Can you think of a rule to find the position of the median given ?
! To find the position of the median for listed data, find : - If a decimal, round up. - If whole, use halfway between this item and the one after.
IQ of L6Ms2 () Frequency ()
Grouped data
Position to use for median:
8.5
! To find the median of grouped data, find , then use linear interpolation.
DO NOT round or adjust it in any way. This is just like at GCSE where, if you had a cumulative frequency graph with 60 items, youβd look across the 30th. Weβll cover linear interpolation in a secβ¦
? ?? ?? ?? ?
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Quickfire Questionsβ¦What position do we use for the median?
Lengths: 3cm, 5cm, 6cm, β¦
Median position: 6th
Lengths: 4m, 8m, 12.4m, β¦
Median position: 12th/13th
Age Freq
Median position: 8.5
Score Freq
Median position: 5
Bro Exam Note: Pretty much every exam question has dealt with median for grouped data, not listed data.
Ages: 5, 7, 7, 8, 9, 10, β¦
Median position: 30th/31st
Score Freq
Median position: 10.5
Weights: 1.2kg, 3.3kg, β¦
Median position: 18th
Volume (ml) Freq
Median position: 6.5
Weights: 4.4kg, 7.6kg, 7.7kgβ¦
Median position: 9th/10th
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Linear Interpolation
Height of tree (m) Freq C.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.
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Linear Interpolation
Height of tree (m) Freq C.F.
The 75th item is within the class interval because 75 is within the first 100 items but not the first 55.
55 100750.6m Med 0.65m
Frequency up until this interval
Frequency at end of this interval
Item number weβre interested in.
Height at start of interval. Height at end of interval.
? ? ?
??
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Linear Interpolation
55 100750.6m Med 0.65m
Frequency up until this interval
Frequency at end of this interval
Item number weβre interested in.
Height at start of interval. Height at end of interval.
Height of tree (m) Freq C.F.
Bro Tip: I like to put the units to avoid getting frequencies confused with values.
π΄ππ πππ=π .π+(ππππΓπ .ππ)
What fraction of the way across the interval are we?
Hence: ?
?
Bro Tip: To quickly get frequency before and after, just look for the two cumulative frequencies that surround the item number.
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Weight of cat (kg) Freq C.F.
More Examples
Median class interval:
10 1816
3kg 4kg
Fraction along interval:
Median:
?
? ? ?
? ?
?
?
Time (s) Freq C.F.
7 2010
12s 14s
Median: (to 3sf)
? ? ?
? ?
?
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Weight of cat to nearest kg Frequency
Whatβs different about the intervals here?
10β12
There are GAPS between intervals!What interval does this actually represent?
9 .5β12.5Lower class boundary Upper class boundary
Class width = 3
?
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Identify the class width
Distance travelled (in m) β¦
Class width = 10?
Time taken (in seconds) β¦
Weight in kg β¦ Speed (in mph) β¦
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?
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Linear Interpolation with gapsJan 2007 Q4
ππππππ=ππ .π+(ππππ Γππ)=ππ .πππ
10297297105111116119120
29 7260
19.5 miles 29.5 miles
? ? ?
? ??
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Test Your Understanding
Age of relic (years) Frequency0-1000 241001-1500 291501-1700 121701-2000 35
Questions should be on a printed sheetβ¦
years
Shark length (cm) Frequency175810
ππππππ=100+( 35Γ200)? ?
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Exercise 2Questions should be on a printed sheetβ¦
π΄ππ πππ=ππ .π+(ππππΓπ)=ππ .π
π΄ππ πππ=ππ+(ππππ Γππ)=ππ .π π΄ππ πππ=ππ+(ππππ Γππ)=ππ .πππ
1
2 3
?
? ?
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Exercise 2
4
?
?
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Quartiles β 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
Items Position of LQ & UQ LQ & UQ
5 2nd & 4th 7
4 1st/2nd & 3rd/4th 6.5 & 12.5
7 2nd & 6th 4 & 9
10 3rd and 8th 3 & 10
Can you think of a rule to find the position of the LQ/UQ given ?
! To find the position of the LQ/UQ for listed data, find or then as before: - If a decimal, round up. - If whole, use halfway between this item and the one after.
IQ of L6Ms2 () Frequency ()
Grouped data
Position to use for LQ:
4.25
! To find the median of grouped data, find and , then use linear interpolation.
Again, DO NOT round this value.
? ?? ?? ?? ?
?
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PercentilesThe LQ, median and UQ give you 25%, 50% and 75% along the data respectively.But we can have any percentage you like.
Item to use for 57th percentile?
24.51?
You will always find these for grouped data in an exam, so never round this position.
Lower Quartile:π1 Median:π2
Upper Quartile:π3 57th Percentile:π57
? ?
? ?
Notation:
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Test Your Understanding
Age of relic (years) Frequency0-1000 241001-1500 291501-1700 121701-2000 35
These are the same as the βTest Your Understandingβ questions on your sheet from before.
Item for :25th
years
years
Shark length (cm) Frequency175810
Item to use for : 14.8?
?
?
?
?
?
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Exercise 3
?
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Exercise 3
π1=ππ+(ππππΓππ)=ππ .ππ1=ππ .π+( π
ππΓππ)=ππ .π?
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What is variance?
πΌπ
πΉππππ’ππππ¦ π·πππ ππ‘π¦
110πΌπ
110
Distribution of IQs in L6Ms4 Distribution of IQs in L6Ms5
Here are the distribution of IQs in two classes. Whatβs the same, and whatβs different?
πΉππππ’ππππ¦ π·πππ ππ‘π¦
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VarianceVariance is how spread out data is.Variance, by definition, is the average squared distance from the mean.
π₯βπ₯()2ππ 2=ΒΏ
Distance from meanβ¦
Squared distance from meanβ¦
Ξ£
Average squared distance from meanβ¦
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Simpler formula for variance
βThe mean of the squares minus the square of the mean (βmsmsmβ)β
Variance
π 2= Ξ£π₯2
πβπ₯2? ?
Standard Deviation
π=βππππππππThe standard deviation can βroughlyβ be thought of as the average distance from the mean.
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Examples
2cm 3cm 3cm 5cm 7cm
Variance cm
Standard Deviationcm
?
?
1, 3
Variance
Standard Deviation
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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PracticeFind the variance and standard deviation of the following sets of data.
24 6Variance = Standard Deviation =
12345Variance = Standard Deviation =
? ?
??
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Extending to frequency/grouped frequency tables
We can just mull over our mnemonic again:
Variance: βThe mean of the squares minus the square of the means (βmsmsmβ)β
ππππππππ=Ξ£ π π₯2
Ξ£ πβ π₯2? ?
Bro Tip: Itβs better to try and memorise the mnemonic than the formula itself β youβll understand whatβs 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.
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Example
May 2013 Q4
π₯=ππππ .ππππ
=ππ .ππππ
We can use our STATS mode to work out the various summations needed. Just input the table as normal.Note that, as the discussion before, on a calculator actually gives you because itβs already taking the frequencies into account.
?
?
?
You ABSOLUTELY must check this with the you can get on the calculator directly.
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Test Your UnderstandingMay 2013 (R) Q3
?
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Most common exam errors
Thinking means . It means the sum of each value squared!
When asked to calculate the mean followed by standard deviation, using a rounded version of the mean in calculating the standard deviation, and hence introducing rounding errors.
Forgetting to square root the variance to get the standard deviation.
ALL these mistakes can be easily spotted if you check your value against ββ in STATS mode.
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Exercises
Page 40 Exercise 3CQ1, 2, 4, 6
Page 44 Exercise 3D Q1, 4, 5
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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?
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Starter
Suppose now after a bout of βstretching you to your limitsβ, youβre 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?)
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The point of coding
Cost of diamond ring (Β£)Β£1010 Β£1020 Β£1030 Β£1040 Β£1050
π¦=π₯β100010
Standard deviation of ():thereforeβ¦
Standard deviation of ():
?
?
We βcodeβ our variable using the following:
New values :
Β£1 Β£2 Β£3 Β£4 Β£5?
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Finding the new mean/variance
Old mean Old Coding New mean New
364 π¦=π₯β20164368 π¦=2 π₯7216354 π¦=3 π₯β208512403 π¦=
π₯22032
1127 π¦=π₯+10379
30025 π¦=π₯β1005 405
? ?
? ?
? ?
? ?
? ?
? ?
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Example Exam Question
f) Coding is
Mean decreases by 5. unaffected.Median decreases by 5.Lower quartile decreases by 5.Interquartile range unaffected.
Suppose weβve worked all these out already.
?
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Exercises
Page 26 Exercise 2E Q3, 4
Page 47 Exercise 3EQ2, 3, 5, 7(Note: answers in book for Q2 are incorrect: therefore
?
?
?
?
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Chapters 2-3 Summary
I have a list of 30 heights in the class. What item do I use for:
β’ ? 8th β’ ? Between 15th and 16th β’ ? 23rd
For the following grouped frequency table, calculate:Height of bear (in metres) Frequency
h=(0.25Γ4 )+ (0.85Γ20 )+β¦
40= 46.75
40=1.17ππ‘π3 π πa) The estimate mean:
b) The estimate median: 0.5+( 1620 Γ0.7 )=1.06πc) The estimate variance:(youβre given ) π 2=67.8125
40β( 46.7540 )
2
=0.329 π‘π3 π π
?
?
?
???
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Chapters 2-3 SummaryWhat is the standard deviation of the following lengths: 1cm, 2cm, 3cm
The mean of a variable is 11 and the variance .The variable is coded using . What is:
a) The mean of ?b) The variance of ?
A variable is coded using .For this new variable , the mean is 15 and the standard deviation 8.What is:
c) The mean of the original data?d) The standard deviation of the original data?
?
??
??