Stats1 Chapter 3 :: Representations of Data
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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 Chapter Overview
We’ve seen so far how data is collected and calculations can be made. We now concentrate on how the processed data can be displayed.
BOX PLOTS AND OUTLIERS
*NEW since GCSE!* Outliers.
CUMULATIVE FREQ DIAGRAMS
HISTOGRAMS
*NEW since GCSE!* Area is not necessarily equal to frequency.
Forming a frequency polygon by joining midpoints.
Changes since the old ‘S1’ syllabus:
Height
Cumulative Frequency
Box Plots allow us to visually represent the distribution of the data.
Minimum | Lower Quartile | Median | Upper Quartile | Maximum |
3 | 15 | 17 | 22 | 27 |
0 5 10 15 20 25 30
Sketch
Sketch
Sketch
Sketch
Sketch
How is the IQR represented in this diagram?
How is the range represented in this diagram?
Sketch
Sketch
IQR
range
Box Plot recap
Interpreting a Box Plot
0 5 10 15 20 25 30
Age (years)
True or false: (click your answer)
“The right box represents more people than the left box.”
False
True
Each box represents 25% of people, i.e. the same number of people!
“The ages are more spread out above the median.”
True
False
The wider the box or whisker, the more spread out the values are within that 25% of the data. We’d say that the data has “positive skew”, but you are not required to know this term.
Outliers
An outlier is: an extreme value.
0 5 10 15 20 25 30
One common definition of an outlier is when we’re 1.5 IQRs beyond the lower and upper quartiles.
(But you will be told in the exam if the rule differs from this)
Outliers beyond this point
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Examples
The diameters of 11 different Roman coins are measured in centimetres:
2.2 2.5 2.7 2.7 2.8 3.0 3.1 3.1 3.2 4.0 4.7
Determine the quartiles and hence any outliers.
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Context: Recall that the standard deviation is, roughly speaking, the average distance of each value from the mean. So the outlier definition is saying we’re at least twice this average distance, which seems like a sensible definition.
In Year 2, you will encounter the normal distribution, which can be used to model data which is clustered about some mean and tails off symmetrical in either direction. If this data was approximately normally distributed, then there is a 5% chance a random observation would fall outside 2 standard deviations within the mean. You will learn then how to make such probability calculations.
Test Your Understanding
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Box Plot Example
Smallest values | Largest values | Lower Quartile | Median | Upper Quartile |
0, 3 | 21, 27 | 8 | 10 | 14 |
0 5 10 15 20 25 30
Draw a box plot to represent the above data.
When there’s an outlier at one end, there’s two allowable places to put the end of the whisker:
Fro Exam Tip: You MUST show your outlier boundary calculations.
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The maximum value not an outlier, 21 (I think this one makes most sense).
OR the outlier boundary, 23.
Use one or the other (not both).
Use a cross for each outlier.
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Test Your Understanding
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c ?
(c) The company claims that for 75% of the months, the amount received per month is greater than £10 000. Comment on this claim, giving a reason for your answer. (2)
£400k £450k £500k £550k £600k £650k £700k £750k
Kingston
Croydon
Box Plot comparing house prices of Croydon and Kingston-upon-Thames:
Comparing Box Plots
“Compare the prices of houses in Croydon with those in Kingston”. (2 marks)
For 1 mark, one of:
For 1 mark:
“The median house price in Kingston was greater than that in Croydon.”
Include some measure of spread.
Include some measure of location (median is best).
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Exercise 3A/3B
Pearson Statistics 1
Pages 32-34, 37
Supplementary Questions:
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(Solutions to (d) and (e) on next slide)
Supplementary Questions:
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Supplementary Questions:
(on your printed sheet)
63
5
52
45
12
17
28
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Time (s) | Frequency | Cum Freq |
9.6 < t ≤ 9.7 | 1 | 1 |
9.7 < t ≤ 9.9 | 4 | 5 |
9.9 < t ≤ 10.05 | 10 | 15 |
10.05 < t ≤ 10.2 | 17 | 32 |
9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3
Time (s)
Cumulative Frequency
32
28
24
20
16
12
8
4
0
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Interquartile Range
= 0.18s
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Plot
Plot
Plot
Plot
These graphs are intended to show the running total of people/things up to a particular value, and are particularly useful in estimating the median and quartiles.
Cumulative Frequency Diagrams
9.5 9.6 9.7 9.8 9.9 10.0 10.1 10.2 10.3
Time (s)
Cumulative Frequency
32
28
24
20
16
12
8
4
0
Estimate how many runners had a time less than 10.15s.
26 runners
Estimate how many runners had a time more than 9.95
32 – 8 = 24 runners
Estimate how many runners had a time between 9.8s and 10s
11 – 3 = 8 runners
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Cumulative Frequency Diagrams
Exercise 3C
Pearson Statistics 1
Pages 39
(Students already confident with cumulative frequency graphs may want to skip this exercise)
Age (years) | Frequency |
| 15 |
| 15 |
10 20 30 40 50
Age
Frequency
15
Pablo is hosting a party. He counts how many people are between 15 and 20, and 20 and 50.
Why is below graph somewhat unhelpful.
How could we fix it?
Click to Start Fro-animation
The 15 people in the second group are more spread out in age, but this graph seems to suggest that people’s ages are spread out uniformly between 15 and 50.
Histograms
Age (years) | Frequency |
15 ≤ a < 20 | 15 |
20 ≤ a < 50 | 15 |
10 20 30 40 50
Age
Estimated Frequency
3
2
1
Let’s presume that within each age group, the ages are evenly spread.
Then there would 3 people of each age in�� the 15-20 group, and 0.5 people of each age in the 20-50 group.
Click to Start Fro-animation
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Frequency Density
The resulting diagram is known as a histogram; it allows us to display the ‘concentration’ (i.e. density) of people per unit value.
The ‘frequency per age’ is known as the ‘frequency density’. In general, given the frequency and class width, we can calculate it using:
Frequency Density =
Frequency
Class Width
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6 7 8 9
Shoe Size
Frequency
Height
1.0m 1.2m 1.4m 1.6m 1.8m
Frequency Density
Bar Charts
Histograms
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Bar Charts vs Histograms
* Not necessarily true. We’ll correct this in a sec.
Use this as a reason whenever you’re asked to justify use of a histogram.
F.D.
Freq
Width
Weight (w kg) | Frequency | Frequency Density |
0 < w ≤ 10 | 40 | 4 |
10 < w ≤ 15 | 6 | 1.2 |
15 < w ≤ 35 | 52 | 2.6 |
35 < w ≤ 45 | 10 | 1 |
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10 20 30 40 50
Height (m)
5
4
3
2
1
Frequency Density
Frequency = 15
Frequency = 30
Frequency = 40
Frequency = 25
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Bar Charts vs Histograms
Still using the incorrect GCSE formula:
Q1
Q2
SKILL #1 :: Area = frequency?
5
4
3
2
1
0
Frequency Density
There were 60 runners in a 100m race. The following histogram represents their times. Determine the number of runners with times above 14s.
9
12
18
Time (s)
Total frequency is known; therefore find total area and hence the ‘scaling’.
Total area = 15 + 9 = 24
Then use this scaling along with the desired area.
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Unlike at GCSE, the area of a bar is not necessarily equal to the frequency; there are just proportional.
A policeman records the speed of the traffic on a busy road with a 30 mph speed limit. He records the speeds of a sample of 450 cars. The histogram in Figure 2 represents the results.
(a) Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample. (4 marks)
M1 A1: Determine what one small square or one large square is worth.
M1 A1: Use this to find number of cars travelling >35mph.
Edexcel S1 May 2012 Q5
7
6
5
4
3
2
1
Fro Tip: We can make the frequency density scale what we like.
Test Your Understanding
(on your printed sheet)
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(b) Estimate the value of the mean speed of the cars in the sample. (3 marks)
M1 M1: Use histogram to construct sum of speeds.
A1 Correct value
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Fro Tip: Whenever you are asked to calculate mean, median or quartiles from a histogram, form a grouped frequency table. Use your scaling factor to work out the frequency of each bar.
Test Your Understanding
(on your printed sheet)
Test Your Understanding
(on your printed sheet)
(c) Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.(2)
(d) Comment on the shape of the distribution. Give a reason for your answer. (2)
(e) State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road. (2)
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(crossed out questions would not appear in new syllabus)
SKILL #2 :: Gaps!
Weight (to nearest kg) | Frequency | F.D. |
1-2 | | |
3-6 | | |
7-9 | | |
2
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1
��0
Frequency Density
1 2 3 4 5 6 7 8 9 10
Time (s)
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Note the gaps affects class width!
Remember the frequency density axis is only correct to scale, so there may be some scaling. However in an exam scaling is unlikely to be required for F.D. if the F.D. scale is already given.
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For simplicity we can set the scaling between area and frequency to be 1.
Jan 2012 Q1
14
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5
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Fro Tip: Be careful that you use the correct class widths!
21 + 45 + 3 = 69
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Test Your Understanding
(on your printed sheet)
SKILL #3 :: Width and height on diagram
An exam favourite is to ask what width and height we’d draw a bar in a drawn histogram.
Q: The frequency table shows some running times. On a histogram the bar for 0-4 seconds is drawn with width 6cm and height 8cm. Find the width and height of the bar for 4-6 seconds.
Time (seconds) | Frequency |
| |
| |
🖉 Bro Tip: Find the scaling for class width to drawn width and frequency density to� drawn height.
Strategy ?
Solution ?
Test Your Understanding
(on your printed sheet)
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SKILL #4 :: Forming a frequency polygon
Recall that a frequency polygon can be drawn by using the midpoint of each interval. This corresponds to the midpoint of the top of each bar in a histogram.
Click to Sketch
Note that the frequency in this interval is 0. That needs to be reflected in the frequency polygon.
Exercise 3D
Pearson Statistics
Pages 43-44
There is a supplementary exercise (available as a separate file for printing) with solutions on the next slides…
Q1
Supplementary Exercise
(on your printed sheet)
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Answer: Distance is continuous
Note that gaps in the class intervals!
4 / 5 = 0.8
19 / 5 = 3.8
53 / 10 = 5.3
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Q2
Supplementary Exercise
(on your printed sheet)
Supplementary Exercise
(on your printed sheet)
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Q3
Supplementary Exercise
(on your printed sheet)
Q4
[June 2007 Q5]
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Supplementary Exercise
(on your printed sheet)
Q5
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Supplementary Exercise
(on your printed sheet)
Q6
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Supplementary Exercise
(on your printed sheet)
Q7
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c ?
d ?
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