Statistics and Probability (Mathematics Curriculum)

Statistics

What Is Statistics?

Statistics is a branch of mathematics that collects, organizes, presents, analyzes and interprets data.

Imagine a teacher wanted to know how well their mathematics class were doing.

They might collect statistics on the scores the class get on a test, where the test is out of 10. That is, they could count how many of the class got 10, how many got 9, how many got 8 etc.

They might want to present this information in an easily understood way, such as in a table or graph.

They might then be interested in finding the typical, or average, score that a student got in that class.

These are all examples of statistics.

Dictionary Definition

The Merriam-Webster dictionary defines statistics as " a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data."

Where Does the Word Statistics Come From?

Statistics comes from the German word 'Statistik' meaning "of the state" (as in country).

Statistics was first used to collect information useful for a government of a country, such as how many people there were of different sexes and ages, what their incomes were etc. This information would have been useful to raise taxes and armies.

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning.

A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Data

Data is information that has been collected or measured.

In this mini-curriculum, you will learn about data and its different types.

Data

Data is a set of facts (such as numbers, measurements or words) that have been collected or measured.

In our example, the data is the test scores of the students that the teacher has collected:

Types of Data

There are different types of data. Some data is in the form of numbers (like a student's test score), others in the form of words (like a student's gender).

Quantitative Data

Quantitative data is data that is described in numbers. It can be counted or measured.

Discrete Data

Discrete data is quantitative data that can only take certain values. It can (often) be counted.

The student's test scores is discrete data. The students can only get a whole number of marks between 0 and 10.

Continuous Data

Continuous data is quantitative data that can take any value (within a range).

The height of the students is continuous data. Heights are not restricted to certain values: such as 5 foot 5 inches or 5 foot 6 inches, but nothing in between.

Qualitative Data

Qualitative data is data that is described in words.

The names of the students is qualitative data.

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Averages

An average is a single value that summarises or represents a set of numbers that are different to each other.

In this mini-curriculum, you will learn about the different types of averages and how to find them.

Average

An average is a single value that is typical for a set of values.

In our example, there will be a single value that is typical of all the student's scores. An average score of 7 means that although students got different scores, 7 is the typical score of the class.

Mean

The mean is an average of a set of numbers.

Find the Mean

To find the mean, add all the numbers together and divide by how many number there are.

Mean of 1, 2, 3 = (1 + 2 + 3) ÷ 3

Mean of 1, 2, 3 = 6 ÷ 3

Mean of 1, 2, 3 = 2

Median

The median is an average of a set of numbers. It is the middle number in a set of numbers that has been arranged in order.

Find the Median

To find the median, order the numbers and find the middle number.

Median of 1, 2, 3 = 1 2 3

Median of 1, 2, 3 = 2

If there are an even number of numbers in the set, there will be two middle numbers. The median is halfway between these two numbers.

Mode

The mode is an average of a set of numbers. It is is the number that appears most often in a set of numbers.

Find the Mode

To find the mode, count how many times each number appears in a set and select the one that appears most often.

Mode of 1, 2, 2, 3 = 2

A set of numbers can have more than one mode.

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Measures of Spread

If the average gives us a typical number from a set of numbers, measures of spread tell us how much the numbers are spread out from each other.

Are they all bunched up close together (like 2, 3, 3, 3, 4)?

Or are they very spread out (like 2, 30, 300, 3000, 4 million)?

In this mini-curriculum, you will learn about different measures of spread.

Range

The range of a set of numbers is the difference between the highest number and the lowest number in the set.

In our example, the range will be the difference between the highest and the lowest score in the class.

Find the Range

To find the range, order the numbers and subtract the lowest number from the highest number in the set.

Range of 1, 2, 3 = 3 − 1

Range of 1, 2, 3 = 2

Quartiles

A quartile is one of three numbers that divide a set into four equal groups.

Lower Quartile

The lower quartile is the middle number between the smallest number and the median. It is the middle number of the lower half of a set of numbers.

Upper Quartile

The upper quartile is the middle number between the median and the highest number. It is the middle number of the upper half of a set of numbers.

Find the Quartiles

There are different methods for finding the quartiles. Different methods give different answers.

Interquartile Range

The interquartile range is the range of the middle half of a set of data. It is the difference between the upper quartile and the lower quartile.

Find the Interquartile Range

To find the interquartile range, subtract the lower quartile from the upper quartile.

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Frequency Tables

Frequency tables are a way of showing data. It lets us see how often a number appears in a set of numbers.

In this mini-curriculum, you will learn about frequency tables and how to find averages and the range from them.

Frequency Table

A frequency table shows how often (how frequently) each number appears in a list of numbers.

Make a Frequency Table

To make a frequency table, count how many times each number appears in a set and record it in a table.

Find the Mean from a Frequency Table

The mean can be found from a frequency table.

Find the Median from a Frequency Table

The median can be found from a frequency table.

Find the Mode from a Frequency Table

The mode can be found from a frequency table. It is the value with the highest frequency.

Find the Range from a Frequency Table

The range can be found from a frequency table. Subtract the lowest value from the highest value.

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Cumulative Frequency Tables

Cumulative frequency tables are frequency tables that keep a running total of the frequencies.

In this mini-curriculum, you will learn about cumulative frequency tables.

Cumulative Frequency Table

A cumulative frequency table is a frequency table with a running total of the frequencies.

Make a Cumulative Frequency Table

To make a cumulative frequency table, first make a frequency table and add a Cumulative frequency column. In this column, add all the entries in the Frequency column from the top row to that row.

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Grouped Frequency Tables

Grouped frequency tables are frequency tables where the numbers are grouped together.

In this mini-curriculum, you will learn about grouped frequency tables.

Grouped Frequency Table

A grouped frequency table is a frequency table where the numbers are grouped together. It shows you how often numbers within each group appear in a list of numbers.

Make a Grouped Frequency Table

To make a grouped frequency table, go through each number in a set and see which of the groups of numbers it belongs in. Count how many times numbers from each group appears and record it in a table.

Find the Mean from a Grouped Frequency Table

The mean can be found from a grouped frequency table.

Find the Median Group from a Grouped Frequency Table

The median group can be found from a grouped frequency table.

Find the Mode Group from a Grouped Frequency Table

The mode group can be found from a grouped frequency table. It is group with the highest frequency.

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Cumulative Grouped Frequency Tables

Cumulative grouped frequency tables are grouped frequency tables that keep a running total of the frequencies.

In this mini-curriculum, you will learn about cumulative grouped frequency tables.

Cumulative Grouped Frequency Table

A cumulative grouped frequency table is a grouped frequency table with a running total of the frequencies.

Make a Cumulative Grouped Frequency Table

To make a cumulative grouped frequency table, first make a grouped frequency table and add a Cumulative frequency column. In this column, add all the entries in the Frequency column from the top row to that row.

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Bar Charts

Bar charts are a way of presenting data. The height of each bar shows how often each value appears in the data.

In this mini-curriculum, you will learn about bar charts.

Bar Chart

A bar chart (or bar graph) is a chart which uses bars to present data. The height of each bar shows how often each value appears in the data.

Read from a Bar Chart

To read from a bar chart, read from the height of each bar the frequency of each value in the data set.

Reading from the bar chart above:

  • 1 appears 1 time.

  • 2 appears 2 times.

  • 3 appears 1 time.

The set of values in our data are: 1, 2, 2, 3.

Create a Bar Chart

To create a bar chart, count how many times each number appears in a set and draw a bar of that height for each number.

Bar Charts and Types of Data

Different bar charts are used fordifferent types of data.

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