Discovering Mathematics

Week 5

BOOK A - Unit 4:

Statistical Summaries

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Unit 4: Statistical Summaries

This unit includes the following sections:

•Questions

•Dealing with data

•Summarising data: Location

•Summarising data: Spread

•Measuring with accuracy and precision•Statistical Pictures (Unit 11)•Discussing TMA

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1. QuestionsThis unit deals with statistical thinking. It starts with questions that can be addressed by statistical methods.

Statistical information is all around you and arises as an attempt to answer questions of various kinds. (e.g. Should people stop smoking?)

Statistical thinking is presented as a helpful way of seeing the world quantitatively. For examples: table of numbers, graphs & charts.

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• Economics– Forecasting

– Demographics

• Sports– Individual & Team

Performance

• Engineering– Construction

– Materials

• Business– Consumer Preferences

– Financial Trends

Section 0: What is Statistics?Application Areas

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Types of statistical questions

• Most of the questions of investigations can be categorized into one of the following types :

- Summarising: how can the information be reduced.

- Comparing: is there a difference?- Seeking a relationship: what sort of

relations is there. 5

Types of statistical questions(II)• Summarizing questions:Summarizing information is by reducing the many figures to just one

representative number or using charts or plots. Examples of investigations:

- How many people die from road accidents each day.- How old are the students studying MU123• Comparing questions:Comparing investigation is to decide whether or not there is a

difference between two things or comparing between two averages. Examples are:

- Do more people, on average, die from road accidents on weekdays or at weekends?

- Are students studying MU123 older or younger than students on LB160?

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Types of statistical questions(III)• Relationship Questions: Investigations could be a relationship between quite separate

things. What the relationship is. For example:- As the numbers of road deaths in different countries linked

to their respective speed limits? - What is the connection between the numbers of hours that

students work on a level 3 course in mathematics and their final grade.

Discuss in the class Activities I and 2 (page 179) and distinguish between the three types of statistical investigations.

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The statistical investigation cycle

There are four stages in most statistical investigations:

P: Posing a question. This is the start of the investigations.C: Collecting data, choosing samples or design a survey.A: Analyze the data, calculating averages and plotting graphsI: Interpreting the results, takes the action back and see whether the data analysis help to answer the posed question.(check Activities 3 and 4 for complete example)

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Activity

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Dealing with Data Data that you collect yourself are called primary data. Data that already exist and can be used or adapted are called Secondary data.It is extremely important when presenting any dataset, whether primary or secondary, is to provide an accurate reference to the data source.Take a quick look at the dataset at page 184. Explain the rows and columns. Solve Activity 6.

Discrete data are data that can take one of a particular set of values. E.g: Number of days in a week you study, or number of websites you visit per day. Binary data: are the data coded by two numerical values. 1,0 Solve activity 9

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Summarising data: location

Statisticians call the location of a dataset to represent a single number that represents an ‘average’, typical or ‘central’ value.

It is always good idea to scan the data, looking closely at the numbers to see if there is a missing data, inspect the data.

Measuring Location:The most important summary of a dataset is a measure of its location. The three most common measures of location are:Mean, the mode and the median.

Let us look at the following examples:

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Summarising data: location (II)

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Summarising data: location (II)

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Summarising data: Spread

A second basic property of data is how widely spread the values are.

Range:TMA marks for six students are: 42, 58, 60, 68, 78, 9242 in MIN and 92 is MAX , this show the data is wide spread.

Another TMA marks are: 60, 65, 72, 74, 75, 80 this ranges from 60 to 80,this is much narrower.

Range is the difference between the two values Maximum and Minimum after arranging the dataset in increasing order.Range = Max – Min = 80-60 = 20

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Summarising data: Spread

Quartiles:Lower quartile (Q1) and upper quartile (Q3)The Median Q2 is (72+74)/2 = 73Lower quartile = 65Upper quartile = 75Interquartile range = 75 – 65 = 10

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Standard Deviation• Standard deviation (SD) is the best known measure of spread. • To find the SD, follow the following steps:

Assume that we have the following data: 1,2,4,6,7

Solve Activity 2616

Summary

• Location and spread are the two key forms of data summary, that two measures of location are the mean and median, and that three measures of spread are the range, the interquartile range and the standard deviation.

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Exercise:consider the dataset below, which consists of the weights inkilograms of 32 women at the start of their pregnancies.54.5 59.1 73.2 41.4 55.5 70.5 76.4 70 52.3 83.2 64.5 49.5 70 52.3 68.2 47.3 66.8 70 56.4 53.6 59.1 44.5 55.9 57.3 69.5 73.2 62.7 73.6 70 56.7 58.2 68.2A class is one of the categoriesThe class frequency is the number of observations in a particular class.

Unit 11. Statistical picturesSection 2: Histograms, bar charts, pie charts

2.1 Histogram (Continuous data)

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Exercise:The bar chart below shows the frequencies with which the six scores 1, 2, 3, 4, 5, 6 cropped up as a result of 30 rolls of a die.

Section 2: Histograms, bar charts, pie charts 2.2 Bar charts (Discrete data)

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Exercise:

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50

100

150

Acct. Econ. Mgmt.

Major

Major Count

Accounting 130

Economics 20

Management 50

Total 200

Section 2: Histograms, bar charts, pie charts 2.2 Bar charts (Categorical data)

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Econ.10%

Mgmt.25%

Acct.65%

36°

Section 2: Histograms, bar charts, pie charts 2.4 Pie Charts (Categorical data)

1. Shows breakdown of total quantity into categories

2. Useful for showing relative differences

3. Angle size =(360°)(percent)

Revision :

The four stages of a statistical investigation cycle PCAIStage 1 Pose a questionStage 2 Collect relevant dataStage 3 Analyse the dataStage 4 Interpret the results

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Revision:

Recall Activity:Here are nine of the common types of task that tend to arise in the C, A and I stages of a statistical investigation. Try to match each task to one of these three stages1.Calculate an average 2.Calculate a percentage 3.Choose a set of values, or sample 4.Make a decision based on an observed, numerical difference 5.Design a questionnaire 6.Draw a conclusion 7.Draw a helpful graph 8.Key the data into a spreadsheet 9.Make a prediction about the real world 23

Discussing the TMA

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Thank you