Module Four: Introduction to Statistics - Mathematics Pathways

Module Four: Introduction to Statistics

Click on the link below to take the Pre-test for Module Four.   You will be given 5 random questions to test your knowledge of Introductory Statistics.  

If you receive more 80% or more, you can move onto the next module.  If your score less than 80%, work through the module lessons and take the post-test at the end of the module.

Make sure you enter your full name and email address so your results can be emailed to you.  You will need to print out or save these results for your records.  You may need to show them to your university.

Module Four:  Pre-test

Application of Statistics in Business

Statistics are used in every walk of life, and are particularly useful in business.

Example

Statistics are widely used in Market Research to help design and market new products. Statistics can be used in analysing the effectiveness of investments and comparing different investment strategies and in predicting the future performance of investments. Statistics are also used in calculating insurance premiums by analysing the likelihood of events and they can be used to estimate risk from weather events for example and the cost of reducing that risk such as by making buildings cyclone proof.

Statistics are also used more broadly in society. They can play a role in predicting changes in a population and demographics to help plan infrastructure and services for the future.  Statistics can also assist you to use information from a small sample to make predictions about the whole population in areas such as health needs, spending patterns, transport use.  Statistics are also used widely in analysing and predicting political outcomes.

The following websites from www.chron.com provide some examples how statistics might be utilised by business in particular:

How Are Statistics Applied in Marketing?

The Importance of Statistics in Management Decision Making

Lesson 1 - Measures of central tendency (Averages)

About This Lesson

Maximum, Minimum, Range and Averages

If you are studying a large set of data such as the daily sales in a large number of branches of a shopping chain or the age distribution of a potential market, it is useful to have ways of describing and summarising the data especially if it contains many items of data.

Watch this short video tutorial from ck-12 to get started

Measures of Central Tendency: A Sample Application

Example

Use the following sample data to help you understand these concepts:

This is the amount spent by 20 customers in a café during breakfast in dollars:

4, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 14, 15, 15, 15, 16, 20

The data has been collected during breakfast and arranged into order.

The main concepts to understand are:

Maximum - the highest value in a set of data

In this case $20

Minimum - the lowest value in a set of data

In this case $4

Range - the difference between the maximum and minimum

In this case 20 - 4 = 16

Averages - there are three kinds of average - the mean, the median and the mode

Mean - (sometimes called The Average or Arithmetic Mean) is the result of adding up all the data items and dividing by the number of items. 

In this case, the sum of 4, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 14, 15, 15, 15, 16, 20

is $215 and there were 20 customers so the mean amount spent is 215 divided by 20 = $10.75

Median - the middle value when all the data items are arranged in ascending order. If there are an even number of data items, it is the mean of the middle two. Sometimes the quarter points are need and these are called quartiles. There is more on that later.

There are 20 items so the median has 10 items on either side

4, 6, 7, 7, 7, 7, 8, 9, 9, 10,     MEDIAN       11, 11, 12, 12, 14, 15, 15, 15, 16, 20

Median is half way between 10 and 11 so is $10.50

Mode - the most commonly occurring data item. 

It is possible for a set of data to have no mode if no item occurs any more frequently than any other or to have more than one mode if there is more than one item that is equally common.

In this case 4, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 14, 15, 15, 15, 16, 20

7 appears with 4 occurrences so the mode is 7.  Notice that if there was one more $15 it would also have 4 so the data would be bimodal with 7 and 15 both being modes.

Test Yourself

This interactive worksheet and self test will help to check your skills and fill in any gaps.

CIMT - Averages

Learn More

Maths is Fun - Finding a Central Value
Maths is Fun - How to Calculate the Mean Value
Maths is Fun - How to Find the Median Value
Maths is Fun - How to Calculate the Mode or Modal Value
Maths is Fun - The Range Statistics
Khan Academy series

 

Practice using this short Self test from ck-12 below. Click on the start button to begin.

 

All Khan Academy content is available for free from www.khanacademy.org  

 

Lesson 2 - Quartiles and box plots

About This Lesson

You have already looked at the median of a set of data, the middle value when all the data items are arranged in ascending order. 

Example 

In the café example from lesson 1:

There are 20 items so the median has 10 items on either side

4, 6, 7, 7, 7, 7, 8, 9, 9, 10,     MEDIAN       11, 11, 12, 12, 14, 15, 15, 15, 16, 20

Sometimes it is useful to look at the quarter points to help describe a set of data. These are called the quartiles. The value with 25% of the data items below it is called the Lower Quartile.

4, 6, 7, 7, 7, LOWER QUARTILE  7, 8, 9, 9, 10,     MEDIAN       11, 11, 12, 12, 14, 15, 15, 15, 16, 20

In this case the lower quartile is half way between the 5th and 6th items which are both 7s so the lower quartile is 7
The value with 25% of the data items above it is called the Upper Quartile.

4, 6, 7, 7, 7, 7, 8, 9, 9, 10,  MEDIAN  11, 11, 12, 12, 14, UPPER QUARTILE 15, 15, 15, 16, 20

In this case the upper quartile is half way between the 15th and 16th items which are 14 and 15 so the upper quartile is 14.5.

Five Number Summary

A set of data can be described using the five number summary which consists of

Minimum 4

Lower Quartile 7

Median 10.5

Upper Quartile 14.5

Maximum 20

This data can also be used to create a Box and Whisker Plot which is a visual representation of the five number summary. Watch the following video from ck-12.
 

http://www.ck12.org/statistics/Box-and-Whisker-Plots/lecture/Data-Analysis-Creating-A-Box-And-Whisker-Plot/

Test Yourself

This interactive worksheet and self test will help to check your skills and fill in any gaps.

CIMT - Box and Whisker Plots (includes Quartiles)

Learn More

Maths is Fun - Quartiles

Khan Academy Series - Box and Whisker Plots which includes the following videos

Test yourself using CK-12 Short Self Test - Box and Whisker Plots.  Click on start to begin.  

Lesson 3 - Frequency Diagrams and Histograms

About This Lesson

Item - Amount Spent	Number of Occurrences
4	1
6	1
7	4
8	1
9	2
10	1
11	2
12	2
14	1
15	3
16	1
20	1
Total	20

Item - Amount Spent	Number of Occurrences
1 to 5	1
6 to 10	9
11 to 15	8
16 to 20	2
Total	20

Learn More

Maths is Fun - Frequency Distribution

Maths is Fun - Pie Chart

Maths is Fun - Calculating the mean from a frequency table

Maths is Fun - Histograms

BBC Bitesize - Frequency Diagram for Grouped Data

(Includes Pie Charts)

Lesson 1 - Cumulative Frequency

About This Lesson

A cumulative frequency graph shows the cumulative totals of a set of values up to each of the points on the graph. It can then be used to find the median and quartiles.

 

Image: http://www.education.gov.uk/sta/professional/b00211213/numeracy/areas/cumulative-frequency

Test Yourself

This interactive worksheet and self test will help to check your skills and fill in any gaps.

CIMT - Cumulative Frequency 

(This tutorial has a few Flash objects to allow you to draw graphs on screen which will not work on some devices. However you can still benefit from looking at it.)

Learn More

Maths is Fun - Cumulative Tables and Graphs

UK Department of Education Tutorial - Cumulative Frequency

Lesson 2 - Stem and Leaf Plots

About This Lesson

A Stem and Leaf Plot is a simple way of summarising data. Watch this short tutorial from ck-12:

Stem-and-Leaf Plots

Once the data is summarised this way it provides a way of finding the range, mode, median and quartiles of the data as you can see in this tutorial:

Learn More

Maths is Fun - Stem and Leaf Plots

YouTube Tutorial - Making a Box and Whisker Plot from a Stem and Leaf Plot 

Lesson 3 - Scatter Plots

About This Lesson

A Scatter Plot is a diagram that represents paired items of data. It is used to test if there is any connection between the two sets of data. For example you would expect to sell more ice cream when the temperature is higher so to test this you would plot Sales against Temperature for a number of days like this:

 

Test Yourself

Check your understanding of this at:

Maths is Fun - Scatter XY Plots

Learn More

If you are not confident, work carefully through the following lessons:

Maths is Fun - Scatter XY Plots

And try the questions at the end.

Khan Academy Series - Scatter Plots including these videos:

 

 

 

 

Practice with the CK-12 Self Test - Scatter Plots.  Click on the start button below to begin.

All Khan Academy content is available for free at www.khanacademy.org 

Reference Sites

Khan Academy Series- Reading and Interpreting Data

Khan Academy Series- Measures of Central Tendency

ck-12 - Statistics

When you have completed the module, check your knowledge with this multiple choice post-test. 

• This test will confirm your level of understanding of the skills in this module.
• Please attempt all questions on the test.
• To achieve 'mastery' (the level at which you are truly competent with these skills) you will need to achieve at least 80%
• You will need to enter your name and email address in the test to have the results emailed to you.

• If you want to use these results to apply to study business at UTAS or UNDA, ensure that you save them to use as evidence that you have passed the module.

Module Four: Post-test