UTAS Home › › Mathematics Pathways › Pathways to Business › 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.
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:
In this topic you will revise the following
Lesson 1 - Measures of central tendency (Averages)
Lesson 2 - Quartiles and box plots
Lesson 3 - Frequency Diagrams and Histograms
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
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
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.
Test Yourself
This interactive worksheet and self test will help to check your skills and fill in any gaps.
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.
About This Lesson
Image: http://www.bbc.co.uk/schools/gcsebitesize/maths/statistics/representingdata1rev3.shtml
A frequency table is a good way to show data especially if there are a lot of items. Using the café example before, writing all the items looks like this
A frequency diagram would look like this
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
It makes it easier to calculate the sum, the mean and makes the mode easy to spot.
When there is a very large amount of data you can use a grouped frequency diagram which puts the data into categories.
Item - Amount Spent Number of Occurrences 1 to 5 1 6 to 10 9 11 to 15 8 16 to 20 2 Total 20
Note: This is only really helpful if there is a very large amount of data. The grouped data can be used to create a pie chart or histogram.
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)
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
About This Lesson
Learn More
Maths is Fun - Stem and Leaf Plots
YouTube Tutorial - Making a Box and Whisker Plot from a Stem and Leaf Plot
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:
And try the questions at the end.Maths is Fun - Scatter XY Plots
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
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.
PLEASE NOTE:
Authorised by the Director, Centre for University Pathways and Partnership
2 May, 2018
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