BishopsGrade 9: Statistics

IntroductionConceptual Foundations

There are five main stages of managing statistical data ….Collect: Collecting data through populations or more frequently using samples. This is

actually a very complicated process, and you will almost always be given previously collected data.

Organise: Tally and frequency tables, stem-and-leaf plotsSummarise: Measures of centre (averages), measures of spread, histograms, box-and-whisker plotsRepresent: Pie charts, bar graphs, histograms, ogives, box-and-whisker plotsInterpret: Use the analysis of the data to interpret the real world situation. The statistical

summary does not provide all the answers but contributes to the decision making.

Learning the statistical methods

Learn the statistical processes using the linked videos and accompanying notes.

Practice your skills with the questions at the end of each lesson.

By the end of this chapter you have to submit:

- The Practise your skills questions at the end of each section, these should be answered on lined paper and handed in as proof that you mastered the skills.

- Task 2b & 4c require you to practice your Excel skills and hand in the file Excel Example on TeacherDashboard.

- Task 3 & 4c will show you how to use Autograph to draw accurate graphs in no time.- While busy with this chapter you will be working on an individual research task based on

provided data sets. You will be expected to summarise the data, represent it and give a short analysis in the form of a written report.

- Submit the original files to all Excel and Autograph summaries and graphical representations, as well as your final report, on TeacherDashboard.

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Task 1: Organising your dataStem-and-leaf plots & Measures of Central Tendency

MEASURES OF CENTRAL TENDENCYThere are THREE measures of central tendency – Mean, Median and Mode.

a) MEAN =

Example: determine the mean of the following data: 4, 7, 15, 8, 9, 7, 13

Mean =

b) MEDIAN The value that lies in the middle of the data set (the value that lies in the middle of all the values)How to find the Median…

Arrange the data set in ascending or descending order. If the data set consists of an uneven number of values, the median will be the

middle value If the data set consists of an even number of values, the median will be the

mean of the two middle values

Example: determine the median of the following data: 4, 7, 15, 8, 9, 7, 13

4 ,7 ,7 ,8 , 9 , 13 ,15

Median=8

Example: determine the median of the following data: 4, 7, 15, 8, 9, 7, 13, 20

4 ,7 ,7 ,8 , 9 , 13 ,15 , 20

Median=8+92

¿8,5

c) MODE is the number that appears the most in the data set. There are three possibilities:

A data set have one modal value (4, 7, 15, 8, 9, 7, 13, 20: Mode = 7)

A data set has more than one modal value 4, 7, 15, 8, 9, 7, 13, 20, 13: Mode = 7 and 13

MEASURES OF SPREAD

RANGE – the range is HIGHEST VALUE minus the LOWEST VALUE of the set of data

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Test your skills

a) The age at which a set of 18 children first owned a cell-phone is listed below:8 12 14 18 12 14 9 9 8

12 13 10 12 13 8 8 4 8

1) What is the mean age?2) Determine the median age?3) What is the modal age?4) Write down the range of ages.

b) The number of cell-phones (in thousands) in different countries is given in the table below:

1. Calculate the mean of this set of data.2. Calculate the median of this set of data.3. Which of the above calculations gives a better representation of the central tendency of

the data? Why?

c) John kept a record of the number of emails he received over a two-week period.The results are shown in the table below.

Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Number of emails

5 13 8 8 7 15 52 12 6 11 12 9 8 10

Using this information:

1) Calculate the mean (or average) number of emails John receives per day 2) Find the mode for the two week period.3) Find the median for the two week period.4) Which of the above “averages” do you think reflects this data best?

Give a reason.

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CountryNumber of cell-phones(in

thousands)

Australia 16 480

China 334 824

Egypt 14 045

France 44 552

Montserrat 1

South Africa 19 500

Russia 74 420

STEM- AND LEAF DIAGRAM and Back-to-back stem and leaf Diagram. The data values are split into tens (10’s) and units. The tens are written under the Stem Column, and the units are written under the leaf column.Example: Represent the following data in a stem – and –leaf diagram.

Speed recorded (in km/h) 80 76 68 65 65 61 60 57 56 52 41

Ensure that you can read the data in ascending/descending order from a stem and leaf diagram. In this example, the values, in ascending order, are read from left to right.

If you turn the plot on its side, it looks like a bar graph.Example of a double stem and leaf diagram/back to back stem and leaf diagram. The data represents the mathematics results (in percentages) of the June and November examination of 25 grade 9 learners.

June September98

3 2 0 7 1 3 35 4 4 0 6 0 1 4 6 6

8 4 4 4 4 3 1 5 0 0 1 1 5 5 98 6 5 5 3 2 0 0 4 2 4 4 6 7 8

6 5 3 2 4 53 2 9

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UNITS (JUNE) STEM (10’S) UNITS (SEPT)

The values for June are ascending from left to right The values for September are ascending from right to left

If you still struggle here are some video clips to help: https://www.youtube.com/watch?v=_7m0Q_m2ppg (3:14)

https://www.youtube.com/watch?v=wCbsKr-PRDQ (18:25)

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Practise your skills

1. The lead concentration (in mg/100ml) in the blood samples taken from 40 blood donors is given below:

a) Construct a stem-and-leaf plot to represent the data

b) Determine the range, mean and median of the data

c) Complete the frequency table given to the right

2. The following shows the speed of 30 cars on the highway (in km/h):

110 118 85 92 93 105 112 100 99 106 87 81 86 84 122 117 102 109 120 121 104 81 83 100 95 93 110 94 97 105

a) Construct a stem-and-leaf plot to represent the datab) Determine the:

i) Rangeii) Medianiii) Modeiv) Mean

c) If the speed limit on this stretch of road is 110km/h, would you say that the average motorist keep to the speed limit?

3. A squad of 20 rugby players was weighed and their masses, in kilograms, were recorded as follows:

81 129 65 114 78 90 99 118 80 78 89 67 105 82 109 88 79 119 95 108

1.1 Construct a stem and leaf diagram

Use the stem and leaf to determine:

1.2 What is the range?

1.3 What is the modal weight?

1.4 What is the median?

1.5 What is the mean (average) weight?

1.6 Which measure of central tendency least reflects the data. Briefly justify

your answer.

.

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Task 2a: Summarising your data manuallyMeasures of Central Tendency with Grouped Data

GROUPED DATA (MEAN; MEDIAN; MODE)

The traffic department investigated where it would be most appropriate to in install speed cameras. As part of their investigation, a survey was done of the different speeds of vehicles on a stretch of a national road. The following table shows the results of the survey:

SPEED (in km/h)

Midpoint of interval (x)

Cumulative Frequency

FREQUENCY(# of vehicles)

f × x

40 < d ¿ 60 50 49 49 2450

60 < d ¿ 80 141 141−49=92

80 < d ¿ 100 275

100 < d ¿ 120 433

120 < d ¿ 140 482

140 < d ¿ 160 499

160 < d ¿ 180 500

ESTIMATED MEAN: When dealing with grouped data, determining an accurate value for the mean is not possible.We refer to an estimate of the mean in grouped data. Determine the midpoint of each interval (we will assume that the midpoint represents the

value of each of the frequencies in that interval (example: midpoint for the first interval is 50. We assume that 49 motorists clocked a speed of 50 km/hr, 92 motorists clocked a speed of 70 km/hr etc.).

Multiply each midpoint with the frequency for that interval. Add these values up and divide by the sum of the frequencies.

MEDIAN AND MODE: will be indicated by the interval

Median = add all the frequencies and divide by 2 to find the position of the median

500/2 = 250 or (500+1)/2 =250,5 . It won’t make a difference with a large number and in an interval

The median will be the 250th piece of data.

The 250th piece of data will appear in the interval 80 < d ¿ 100. (look at the cumulative frequency there are 275 pieces of data less than 100

Therefore the median interval is 80 < d ¿ 100.

We talk about a modal class (where the biggest frequency appears).

In the above example, it will be 100 < d ¿ 120.

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Extension – How to work out the ESTIMATED MEAN on your calculator There are 3 STAT keys on the calculator

You find the first stat key by pressing MODE. The STAT key will be one of the options on your calculator.

You can find the second stat key by pressing shift mode, scroll down, and the stat key should be one of the options –this stat key is used to set your calculator on frequency on or off (normally used to calculate the mean of grouped data)

The other stat key is found above the 1 on your calculator. To activate this key, press SHIFT 1.

STEP 1: Press shift mode. Scroll down and press the stat key

STEP 2: Choose frequency ON

STEP 3: Press Mode and choose the STAT function

STEP 4: Choose the VAR (VARIANCE) function

STEP 5: enter the midpoints in the x column and the frequencies in the frequency column, pressing = after every entry and the AC key once all entries have been completed.

STEP 6: Press SHIFT 1. Choose the VAR (VARIANCE) function.

STEP 7: n = number of data entered, = mean, . Choose and press =.

If you still struggle here are some video clips to help:

https://www.youtube.com/watch?v=Z1R3UEJk0TE (14:20)

https://www.youtube.com/watch?v=z_CBPRrEJUw (2:52)

Practise your skills

1.

The yields, to the nearest kg, of 140 fruit trees given in this table.Calculate the mean estimate

Yield (kg) Number of trees

10 x < 15 4

15 x < 20 8

20 x < 25 13

25 x < 30 28

30 x < 35 39

35 x < 40 30

40 x < 45 11

45 x < 50 7

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2. The graph shows the number of hoursa sample of people spent watching

TV over the course of a weekduring the summer holidays.

a) Use the graph to complete the frequency table for the sample

TV time(h hours)

Frequency (f)

Midpoint (M)

f × M

0 h < 10 13 5 6510 h < 20 2720 h < 3030 h < 40 3540 h < 3050 h < 60

TOTAL n =

b) Calculate the estimate mean for the summer viewing time for this samplec) In which interval would you find the median?

d) Another survey is carried out during the winter. Give one difference you expect to see in the data.

(Source: SEG O-level exam)

Task 2b: Summarising your data in Excel

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Measures of Centre -- Mean of grouped data

Excel can help us to manage much bigger datasets that we can manage manually. In this task, you are going to learn how to create a frequency table in Excel.

Download the file Gr09_Excel_Example from the intranet. If you open it you will find a dataset containing

Reaction times when pushing a stopwatch using the left and right hand.

Next to the dataset is a frequency table. The frequency table for their left hand is already done, you have to repeat the process to create the frequency table for their right hand.

How to create frequency tables and calculate estimated mean on Excel:

Open the file and look at the example. Double click on any cell to see the formula that was used to generate that result.

If you hover over an orange cell, a comment will appear telling you what to do.

If you still cannot get the formulas right, watch this video.

https://youtu.be/_3yxfGv-94w

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APPLY to your ASSIGNMENTUse the skills you have just practiced and create a

frequency table for the data sets you have selected.

Remember:

You should have 6 – 12 intervals/bins There can be no gaps between the intervals There can be no overlapping between the

intervals.

Task 3: Representing your dataHistograms and bar graphs

Introduction

Graphing (representing) data is the first and often most important step in data analysis. A well-chosen graph with an appropriate scale on the axes communicates vast amounts of information about the data, information that is often hidden by the details of the raw data.

Bar graphs VS Histograms

Bar graph Histogram

Spaces separate bars Bars are drawn next to each other. Spaces only occur when there are no items in that class.

The x-axis is notated with categories. e.g. Types of cars, cities, number intervals

The x-axis is a continuous number line drawn to scale.

Bars are of equal width. Bars can have varying widths.However, we are only going to work with histograms that have equal bin widths.

Figure 1. Bar graph of ages attending a leadership workshop

Figure 2. Histogram of ages attending a leadership workshop

The first class interval includes ages 20 x < 30. That is 20 is included, but 30 is in the next interval. The width of the column in a histogram is called the class interval or bin width.

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Frequency

Age

Notes on histograms

Discovering Algebra (pg 59)

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Practise your skills

1. The yields, to the nearest kg, of 140 fruit trees are tabulated on the right.

Construct a histogram to represent the data in a summarised form.

Yield (kg) Number of trees

10 x < 15 4

15 x < 20 8

20 x < 25 13

25 x < 30 28

30 x < 35 39

35 x < 40 30

40 x < 45 11

45 x < 50 7

2. The ages of drivers involved in fatal accidents over a one week period are given below.

Using a suitable bin width, create a frequency table and then draw a histogram for the data

3.

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Task 4a: More summarising your dataMeasures of Spread -- Box-and-whisker plot

Introduction

The 5 number summary is used to divide a set of data into 4 equal parts or quarters. We start by finding the middle value (median) which divides the data set into 2 equal parts. Then we find the lower quarter point (lower quartile) and the bigger quarter point (upper quartile). We obviously also need to indicate the minimum and maximum values, and these make up the 5 numbers used to summarise the set of data.

This summary can then be represented on a graph. We represent the 5 number summary on a number line as a ‘box’ for the middle 50% of the data and a ‘whisker’ on each end showing the bigger quarter and lower quarter respectively.

While you will be required to find the 5 number summary in small samples (e.g. 20 items) to test if you have learned the techniques, in reality, we only use this summarising tool for larger (e.g. > 100 items) samples.

Self-paced learningGo to https://www.brainingcamp.com/lessons/box-and-whisker-plots/

work through the lessons, questions, problems and challenges.

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APPLY to you ASSIGNMENT

Create a histogram of your datasets using Autograph.

Use the data in the frequency table you created yesterday.

https://drive.google.com/file/d/1Ahgd9gZzu116VLACPJYliIMBRIOfvtOj/view (2:19)

Self study

Task 4b: More summarising your dataMeasures of Spread -- Box-and-whisker plot

Understanding the basics of the 5-number summary (in a small sample)

Example 1: The median is included in the dataset.

Step 1: Place the data in order from smallest to largest

2,5 3 4,2 4,8 5 5,3 5,3 5,7 6,1 6,8 7 7,2 9,8

Step 2: Find the middle value, called the median, of the data.

Median (also called the 50th percentile or 2nd quartile)

2,5 3 4,2 4,8 5 5,3 5,3 5,7 6,1 6,8 7 7,2 9,8

The median divides the data set into two halves.

5,32,5 3 4,2 4,8 5 5,3 5,7 6,1 6,8 7 7,2 9,8

Step 3: What number (lower quartile) divides the lower half of the data into two quarters?

What number (upper quartile) divides the upper half of the data into two quarters.

Lower quartile Median Upper QuartileQ1 Med (or Q2) Q3

4,5 5,3 6,9

2,5 3 4,2 4,8 5 5,3 5,7 6,1 6,8 7 7,2 9,8

Notice: The data set is divided into quarters (with each quarter having 3 items)In this case, the lower and upper quartiles are not elements of the data set. So we use 4,5 which is midway between 4,2 and 4,8 and divides the bottom half into two quarters.

Step 4: Place the summary as short vertical lines on a number line.

Min Q1 Med Q3 Max

Putting in the box and the whiskers a box-and-whisker plot (also called a box plot)

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Example 2: The median is NOT included in the dataset.

Step 1: Place the data in order from smallest to largest

2,5 3 4,2 4,8 5 5,3 5,3 5,7 6,1 6,8 7 7,2 7,3 9,8

Step 2: Find the value, called the median, that divides the data set in half. In this case, the median is NOT one of the items in the dataset. Hence, choose the value (5,5) that is halfway between the two middle values (5,3 and 5,7).

Median

5,52,5 3 4,2 4,8 5 5,3 5,3 5,7 6,1 6,8 7 7,2 7,3 9,8

Lower half Upper half

Step 3: What number (lower quartile) divides the lower half of the data into two quarters?

What number (upper quartile) divides the upper half of the data into two quarters.

Lower quartile Median Upper QuartileQ1 Med (or Q2) Q3

5,52,5 3 4,2 4,8 5 5,3 5,3 5,7 6,1 6,8 7 7,2 7,3 9,8

Notice: The data set is divided into quarters (with each quarter having 3 items)

5,52,5 3 4,2 4,8 5 5,3 5,3 5,7 6,1 6,8 7 7,2 7,3 9,8

Step 4: Place the summary in a box plot on a number line.

Min Q1 Med Q3 Max

Practise your skills

1.

a) Construct a stem-and-leaf diagram to represent these datab) Draw a box-and-whisker plot to represent these datac) Give one advantage of using (a) and (b) to illustrate data such as given above

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Steps to finding the 5 number summary (in a large sample)

When there are too many items to write them out sensibly in an ordered row, we find the position of

the median by looking for the item in the list, where n

is the frequency.

The lower quartile is the value of the item. That is, look for the value of the item in the list.

The upper quartile is the value of the itemExample The stem-and-leaf plot represents the IQ of 64 students. Determine the 5-number summary for the set of data and represent the summary in a box plot.

Freq1 6 80 72 7 5 81 8 38 8 5 5 5 6 6 7 7 8

11 9 0 1 1 1 1 3 3 3 4 4 4 9 9 6 6 7 8 8 9 9 9 9 7 10 0 1 1 2 2 4 4

13 10 5 5 5 6 6 6 7 7 7 7 7 8 9 4 11 0 0 1 4 3 11 6 6 7 2 12 2 3 1 12 8

131 13 61 14 1

Stem unit = 10Leaf unit = 1

Median: = 32,5th item. There are 32 items up to the last value of 99. The median is halfway between 99 and 100, that is, 99,5

Lower quartile: = 16,25th item. There are 12 items up to the value 88. 91 is the 16th item, and 91 is also the 17th item. Hence, 91 is the 16,25th item.

Upper quartile: 48,75th item. Counting down to the 48,75th item we find 107

Box plot:

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Practise your skills

1. The stem-and-leaf plot represents the maximum temperature load of industrial bearings. The values cluster uniformly in a range of 1530 to 1543 degrees, then disperse gradually at higher temperatures. Summarise this information in a box plot

http://www-01.ibm.com/support/knowledgecenter/SSLVMB_20.0.0/com.ibm.spss.statistics.cs/explore_normality.htm

2.

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(IEB 2013)

3. During soccer season, Albert, Bettina and Paul sold cold drinks at three different mobile tuck shops. They kept a record of how many drinks they sold at each game. A summary of the sales is shown in the box plots below. They want to use the data to find the best location for the mobile tuck shop.

a. Who had the highest median?

b. Who sold the least number of soft drinks at a game?

c. Who had the highest proportion of sales above 30 drinks per game?

d. Who had 75% of his or her sales above 70 drinks per game?

e. What information about the best location for the mobile tuck shop can you get from these plots?

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Outliers

Mathematics Enhancement Programme: GCSE Statistics (pg 39)

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Example

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Task 4c: More summarising your dataMeasures of Spread -- Box-and-whisker plot

Like with frequency tables and histograms, between Excel and Autograph, we can easily create box-and-whisker graphs of big data sets.

Go back to the file Excel_Example and click on the second tab called 5 number summaries.

Next, to the dataset, you will find the 5 number summary for left hands. Hover over each cell to see what formula was used and repeat it to calculate the 5 number summary for the right hand.

Click here to see how to calculate your 5 number summary in Excel https://drive.google.com/file/d/1B84F68iTfrmbbNOtOtHGvZ9LLGmDgoeL/view

Your teacher will tell you what the correct answers should be.

Now transfer the data to Autograph and draw a Box-and-Whisker graph. Take a screenshot of this and paste it into the spreadsheet.

Drawing a Box-and-whisker plot from the 5-number summary: https://drive.google.com/open?

id=1IJ15JVA38wU0bxq2ifg9uX4BFr1ecMwI

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APPLY to your ASSIGNMENT

Use the skills you have just practiced and create Box-and-whisker plots of the data sets you have selected to use in your report.

Remember:

You should be able to read the 5-number summary from the graphs. There should be a separate Box-and-whisker plot for each data set.

They should be drawn on the same axis to enable to do comparisons.

Now you are ready to start writing your report.

Last lesson: Recalculating the meanExample

After 5 class tests, Joan had a mean maths score of 58,4%. What does she need to score in the 6th, and last, test to achieve a term mark of 60%?

Total after 5 tests = 5 × 58,4 = 292To get an average of 60% Joan needs 6 × 60 = 360 total score after 6 testsThat is, Joan needs 360 – 292 = 68% for the last test

[Hint: Always determine the aggregate (total) before you recalculate the mean]

Exercise

Mathematics Enhancement Programme (CIMT)

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To do checklist

Do I understand the terminology?

- Mean, median & modal class- Range- Quartiles- Five number summary- Inter-quartile range (IQR)- Box-and-whisker plot- Outlier

- Frequency table- Grouped data or class interval - Group limit or class boundary- Bin width- Stem-and-leaf plot- Bar graph; pie chart; histogram

Can I

Organise data in a stem and leaf plot.

Organise data into bins of a suitable interval in a frequency table

Recognise the modal class

Calculate the mean estimate of grouped data on paper

Calculate the mean of a large set of data using Excel

Determine the 5 number summary of a large set of data in Excel

Draw a histogram by hand and using Autograph

Draw a box-and-whisker by hand and using autograph

Have I

Chosen a set of data to analyse

Asked (research) question(s) related to the data that I will try to answer

Organised the data into a frequency table & histogram on Excel / Autograph

Summarised the data for measures of centre and spread using a 5 number summary and box-and-whisker graph.

Interpreted the data to answer my questions and write a report.

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