Summary Statistics – Mean, Median, Mode and Range

Learning Intention: Students will understand the meanings of the terms “mean”, “median”, “mode” and “range” and be able to calculate these summary statistics from data supplied to them.

Success Criteria: Students will be able to calculate the mean, median, mode and range from given data.

We are now doing a unit of work on statistics and you will need to know the following terms:

  • Categorical data (cannot be measured by quantity, for example, colour of eyes, favourite footy team)
  • Numerical data (can be measured by quantity, for example, number of students at school)
  • Continuous data (eg. height, weight, length of foot) can be a decimal number.
  • Discrete data  (eg. number of pages in a book, number of students in a class) cannot be a fraction or decimal.
  • Mean = average (total of all scores divided by number of scores)
  • Median = middle score (arrange data from smallest to largest and find the middle number)
  • Mode = most frequent score
  • Range = difference between largest and smallest score

The Australian Bureau of Statistics collects data to assist businesses and the government to plan for the future – where to build freeways, hospitals and schools, for example. Every four years they distribute a census across Australia to gather data about housing and population. How is a “survey” different to a “census”? Each year they invite schools to particpate in the “CensusAtSchool” program, which I have signed us up to. After you have completed the survey, we can use the data to work out the summary statistics for the “hours of sleep per night“. There are some other great activities we can do at quick maths activities from the census at school survey.

 

 

Student surveys, data and graphs

Learning Intention: Students will understand how to collect data using a tally and create a frequency table and bar graph using the data. They will understand how to convert fractions to decimals and percentages. They will create a pie chart using this data by converting 100% to 360 degrees.

Success Criteria: Each student will produce a poster that includes a frequency table (including fractions, decimals and percentages), bar graph and pie chart of their chosen data, collectecd from the Year 7 Maths Survey.

  1. First collect your data in tally form
  2. Add each category and find the total
  3. Represent each category as a fraction
  4. Convert to a decimal (2/25 = 16/100)
  5. Convert to a percentage 16/100 = 16%
  6. Create a bar graph using this data
  7. Remember to add SALT to your graph – Scale, Axes, Labels, Title
  8. Turn your bar chart into a pie chart (multiply percentage by 3.6 because 100% = 360 degrees)
  9. Make sure you have a key to interpret your data.
  10. Add a beautiful title and colour to present your poster.

Go to Create-A-Graph and use your data to check that your graphs are correct. Print out the computer generated graphs to add to your poster.

 

 

Gradient

The gradient of a ramp is very important if you are a builder or in a wheelchair – too steep and it is too difficult to wheel up and too shallow and it is very long and expensive to build. The Australian Standard (AS1428.1) requires that
ramps should be of a gradient of 1:14 (if over 1250mm in length) and 1:8 if less than 1250mm in length. The ramps at school were built 15 years ago; measure them and determine if they meet the current Australian Standards.

Take a photograph and measure the slope of the slide (or another example of gradient) in the playground. What is it’s gradient? (rise divided by run). Label your image (in Paint) and send it to my email address. The following screenshot shows the slopes generated in Graphmatica (Free download here). The pink line is the slope of the ramps inside our old school building. It sits between the recommended Australian standard (white) and the maximum Australian Standard (red). The new building has a wooden ramp with a slope of 1:14, which is the recommended Australian standard. Choose one of the slopes we measured and describe it in the comments below.

 

Have you got SALT on your graph?

Learning Intention: Students will learn to draw a line graph to represent a data set, including the appropriate scale, axes, labels and title. They will also use technology to create a graph using the same data to compare the process and the product.

Success Criteria: A successful line graph will include the following:

  • lines drawn neatly with a ruler and greylead pencil or a digital graph with appropriate data
  • an appropriate scale to show the data clearly
  • clearly labelled and equal increments on both the horizontal and vertical axes
  • labels on each of the axes that identifies the appropriate data (time in years, population in 100,000’s for example)
  • a clear and accurate title that explains the purpose of the graph
  • Students may also be able to extrapolate the graph to make a prediction about future data.

“Every five years the Australian Bureau of Statistics (ABS) runs the Census of Population and Housing. This year 29 000 collectors will be part of a 43 000 strong Census workforce that will paint our national portrait in numbers. The Census is a questionnaire filled out by everyone who is in Australia on Census night, except foreign diplomats and their families. It’s so important that it’s mentioned in the Australian Constitution.

The Census counts the number of people in Australia, and information about them like what work they do, what education they have and the households they live in. This information helps decide where services such as hospitals, schools and roads will be built. The Census of Population and Housing is also used as the starting point to estimate the population of Australia, the states and territories and small communities.

This Census involves delivering 14.2 million Census forms to Australia’s 9.8 million households and then transporting and processing more than 46 million pages of data. Census is also changing with the times: 30% of the population are expected to fill out their forms online using eCensus.” Read more about the Census at “Maths by Email”.

The following task uses data taken from the results of the census to produce a line graph that shows the changes in the “Estimates of the Indigenous Australian Population since 1901”. Complete the graph with greylead and a ruler and answer the following questions:

  1. Give reasons for why a line graph is the most appropriate way to present this data.
  2. Explain why a histogram is an incorrect way to present this data.
  3. Look at the shape of the line in your graph. What sort of graph is this?
  4. Use extrapolation to estimate the Indigenous population for 2011.
  5. Do you think your estimate will be correct? Give reasons for your answer.

Now use Create-a-Graph, Excel, ChartGo or the Online Chart Tool to create another line graph using the same data. Compare your paper version with the digital version. In the comments section below let me know which tool you used and how the graphs compared. Which was easier and why? Which was a better product and why? Which tools would you prefer to use? How might you use these tools in the work place?


This is James’ graph, which includes SCALE, AXES, LABELS and a TITLE. Well done!

Clothing Combinations and Dinner Menus

clothing_combinations

Problem 1: Imagine you had three different pairs of pants in your wardrobe and four different tops. How many different combinations could you wear? How can you work it out for any number of articles of clothing? Now add two hats. How many outfits could you put together? To make the most out of your wardrobe (that is, most combinations) would you add another pair of pants, top or hat?

Problem 2: Three bus captains are to be chosen by putting 11 names in a hat – How many different combinations are possible?

Problem 3: At a buffet dinner, you have a choice of two soups (pumpkin or spring vegetable), four different main courses (chicken, lamb, beef or vegetarian) and three desserts (fruit salad, chocolate mousse or cheesecake). How many different combinations of three course meals could there be?

Problem 4: Your “combination” lock on your locker (really should be called a “permutation” lock, because the order of the numbers does matter!) has three numbers, each from zero to 39. What is the total number of possible lock combinations?

Problem 5: In a group of school students, 26 people barrack for Geelong and 32 like playing cricket. How many people might there be in the group? (You may like to use a Venn diagram or a table to help with this question. – Thanks to Peter Sullivan for this question).

Problem 6: At a party there were 50 people; 35 ate some of the fish, 30 ate some of the chicken dish and 5 ate nothing. Draw a diagram to show this information. Work out:

  • what fraction of the people at the party ate only fish?
  • what fraction of the people at the party ate only chicken?
  • what fraction of the people at the party ate both fish and chicken?
  • what fraction of the people at the party did not eat chicken?

More about Combinations and Permutations from Dr. Maths at the Maths Forum.

And even more about Combinations and Permutations from Maths is Fun!

Write a comment below to tell me the answer to the problem you liked best and tell me why you thought that was the most interesting problem.

Using an iPod for probability experiments

iPod image

Screenshot from iChoose

Year 6/7 students continue to explore probability in our maths classes, today with iPod touches. “iChoose” is a free application, with a number of options for random choice results – coin toss, dice, cards, girl/boy and more.  Students did  timed trials to collect data (both coin tosses and dice throws) and collated class results. Each student used the iPod calculator to work out their percentages of each result and then compared individual data with class data to show that the greater the number of trials, the closer to the theoretical probability the experimental data gets.

We also used real dice and coins (swapping over so all students had a turn at both the real and virtual) and compared the results. At the end of the lesson, students completed an ‘exit slip’, with three important things that they have learnt about probability. Most students were able to complete this successfully, writing how to calculate probability, how all outcomes add to 100% and about experimental and theoretical probability. Even students with greater learning needs were able to state that they had learnt how to work out percentage and how to tally results. It is pleasing when you can engage students with a wide range of learning abilities in one class and hear that each of them has benefited from the lesson.

Column and Bar Graphs from Surveys

6-7RFavouriteTV shows

Create-A-Graph image by 6/7R student

Last lesson each of the students in both 6/7 classes completed a survey (created in Google Docs and embedded on the page with the tab “Survey2” at the top of this page) and the data was saved in a Google Docs spreadsheet. Then the students chose one of the data sets (favourite ice-cream flavours, footy teams, TV shows etc) and produced a bar chart or column graph using the class results. You can see from the graph above that “The Simpsons” is one of the most popular TV shows amongst 12 and 13 year olds at Hawkesdale P12 College!

We will use the same data to create a stem and leaf table with height data and  a scatter plot with height and foot size. Students have also been asked to collect a graph from a newspaper or magazine and will describe what the graph shows. This Voicethread, Year 6/7R Student Data and Graphs, contains some of the graphs produced in class.

Creating Graphs

salt_shakers4

Image Source

This week in 6/7 Maths we will use out data to create bar charts, line graphs, pie charts, stem-and-leaf plots and scatter graphs. The Kids Zone has a great tool for creating different types of graphs at Create-A-GraphChartGo is another on-line tool for creating graphs. Think about which graphs are best for each purpose and the information you are trying to convey.

Choose which data you would like to graph and decide which type of graph to use. Time/Temperature graphs are usually line graphs because the data is continuous. Discrete data (favourite colour, footy teams or fast foods for example) are better represented using bar or column graphs. When you are graphing percentages of a population a pie chart is most suitable. The other thing you need to remember is that all graphs need SALT on them. Make sure you season your graphs well with the following information:

S = Scale

A= Axes

L = Labels

T = Title

Surveys and Data

pie_chart_hours_in_a_day

Image Source

While I am away next week (Wednesday, Thursday and Friday) you are required to complete a class survey. Choose a topic of interest (favourite AFL football team, favourite ice-cream flavour, favourite sport etc), with at least four options, plus “Other”.

  • Tally your class results in a table.
  • Convert the results to percentages (use a calculator for this if necessary)
  • On a 100cm strip of paper, colour each section corresponding to the data collected. For example, if 10 out of 25 people voted for Essendon, colour 40cm of the strip in essendon colours.
  • Tape the ends of the paper strip together and create a pie chart on A3 paper from the paper circle.
  • Colour your pie chart carefully and give it a heading. Make sure you include a key to interpret the results.

Please also complete “Survey 2” by going to the tabs at the top of this post.