Lesson Objective Understand the difference between a Bar Chart and a Frequency Chart Be able to draw Frequency Charts using sensible groupings Extend to look at Frequency Polygons

Distance, d, from school in Miles Draw Bar Charts for both of these sets of data: Students were asked how many brothers or sisters they had Students were asked how far from school they lived Number of Siblings Number of Students 2 1 5 4 3 Distance, d, from school in Miles Number of Students 0 ≤ t < 1 5 1≤ t < 3 6 3 ≤ t < 5 5 ≤ t < 10 4 10 ≤ t < 20 2

Distance, d, from school in Miles Draw Bar Charts for both of these sets of data: Students were asked how many brothers or sisters they had Students were asked how far from school they lived Number of Siblings Number of Students 2 1 5 4 3 Distance, d, from school in Miles Number of Students 0 ≤ t < 1 5 1≤ t < 3 6 3 ≤ t < 5 5 ≤ t < 10 4 10 ≤ t < 20 2 Frequency 0 ≤ t< 1 1 ≤ t<3 3 ≤ t<5 5≤ t< 10 10 ≤ t< 20 1 2 3 4 5 Distance in miles Frequency 6 0 5 10 15 20 1 2 3 4 5 Frequency 6 0 1 2 3 4 1 2 3 4 5 Number of Siblings

Distance, d, from school in Miles Draw Bar Charts for both of these sets of data: Students were asked how many brothers or sisters they had Students were asked how far from school they lived Number of Siblings Number of Students 2 1 5 4 3 Distance, d, from school in Miles Number of Students 0 ≤ t < 1 5 1≤ t < 3 6 3 ≤ t < 5 5 ≤ t < 10 4 10 ≤ t < 20 2 Frequency 0 5 10 15 20 1 2 3 4 5 Frequency 6 0 1 2 3 4 1 2 3 4 5 Number of Siblings

Bar Charts: Used for discrete data or categorical data Bars are always equal widths Labels for each bar go under the bar Frequency Charts: Use for grouped data (usually continuous data) Widths of the bars shows the widths of the group The horizontal axis is a continuous scale/number line

Why do we group continuous data? How do we choose the best groupings?

Draw a Frequency Chart for this data. People on a bus route were asked how long they had waited to get on the bus. The times are shown below (in minutes) 0 0 0 0 0 0.5 1.2 1.9 2.7 2.9 3 3.5 4.2 5.6 8.1 8.2 9 11.7 12.9 14.7 15.5 17.8 23 31 Draw a Frequency Chart for this data. Think carefully about your groupings!

Intervals of every 2

Interval of 5

Interval of 10

Mixed Intervals

0 0 0 0 0 0.5 1.2 1.9 2.7 2.9 3 3.5 4.2 5.6 8.1 8.2 9 11.7 12.9 14.7 15.5 17.8 23 31

0 0 0 0 0 0.5 1.2 1.9 2.7 2.9 3 3.5 4.2 5.6 8.1 8.2 9 11.7 12.9 15.5 17.8 23 31 14.7

http://illuminations.nctm.org/ActivityDetail.aspx?ID=78

Key points: Draw a Bar Chart for Discrete Data Draw a Frequency Chart for continuous data When grouping data: Look for the smallest and biggest values Divide the gap between the biggest and smallest values into roughly equal chunks (2, 5, 10 etc) Try to avoid a jagged graph by merging groups together Whilst not ideal – it is okay to have different sized intervals. (This is sometimes essential to include ‘outliers’) If you join the midpoints of a frequency chart at the tops of the bars you get a ‘Frequency Polygon’. You can draw this without drawing the bars if asked to do so.

Draw a Frequency Polygon for this data. Pupils at a Primary School in Year 6 were asked how long it took them to journey to school. The results are shown below: Time in mins, t Number of Pupils 0 ≤ t < 10 3 10 ≤ t < 15 7 15 ≤ t < 20 8 20 ≤ t < 25 4 30 ≤ t < 40 1 40 Draw a Frequency Polygon for this data.

Draw a Frequency Polygon for this data. Pupils at a Primary School in Year 6 were asked how long it took them to journey to school. The results are shown below: Time in mins, t Number of Pupils 0 ≤ t < 10 3 10 ≤ t < 15 7 15 ≤ t < 20 8 20 ≤ t < 25 4 30 ≤ t < 40 1 40 Draw a Frequency Polygon for this data.

Lesson Objective Consolidate knowledge of Frequency Charts Be able to draw Frequency Polygons What are the main differences between a Frequency Chart and a Bar Chart? What is a Frequency Polygon?

Key points: Draw a Bar Chart for Discrete Data Draw a Frequency Chart for continuous data When grouping data: Look for the smallest and biggest values Divide the gap between the biggest and smallest values into roughly equal chunks (2, 5, 10 etc) Try to avoid a jagged graph by merging groups together Whilst not ideal – it is okay to have different sized intervals. (This is sometimes essential to include ‘outliers’) If you join the midpoints of a frequency chart at the tops of the bars you get a ‘Frequency Polygon’. You can draw this without drawing the bars if asked to do so.

Exam Questions from AQA

People are really bad at drawing Frequency Polygons What is wrong with these frequency polygons?.

People are really bad at drawing Frequency Polygons What is wrong with these frequency polygons?. Obviously we don’t need the bars but that isn’t the real issue. The scale along the bottom is not continuous (a number line). This is a Bar Chart and doesn’t have a continuous scale along the x-axis. The data is discrete so a Frequency Polygon is pointless as the data cannot take values between 3 and 4 so showing the shape of the data between 3 and 4 is silly

Lesson Objective Understand the drawbacks of a Frequency Chart for plotting Continuous Data Be able to alter a Frequency Chart to become a Histogram

I observed how long people had to wait (in minutes) to catch the bus at a Park & Ride site. I drew three graphs to illustrate my data. 30 25 20 15 10 5 0 to 1 1 to 5 5 to 10 10 to 20 20 to 40 Frequency Time (mins) Frequency Time (mins) 30 25 20 15 10 5 40

Frequency Time (mins) Frequency Time (mins) 30 25 20 15 10 5 0 to 1 0 to 1 1 to 5 5 to 10 10 to 20 20 to 40 Frequency Time (mins) Frequency Time (mins) 30 25 20 15 10 5 40

Good: You can easily tell how many people are in each interval. 30 25 20 15 10 5 0 to 1 1 to 5 5 to 10 10 to 20 20 to 40 Frequency Time (mins) Good: You can easily tell how many people are in each interval. Poor: Interval widths look the same even though they are not because the bars are all the same width. Good: You can quickly see the proportions for each interval; for example around half waited less than 5 mins Poor: No idea of how many people are actually in each interval. Frequency Time (mins) 30 25 20 15 10 5 40 Good: You can easily tell how many people are in each interval. The differing sizes of each interval are obvious. Poor: The widest bars ‘catch the eye’ and suggest a higher frequency/proportion than they actually have.

We like the frequency chart because: It shows the unequal interval widths clearly It shows the frequency BUT What about the proportions? Frequency Time (mins) 30 25 20 15 10 5 40

÷ 4 ÷ 5 ÷ 10 ÷ 20 Frequency Time (mins) Frequency Density Time (mins) 30 25 20 15 10 5 40 Time (mins) ÷ 4 ÷ 5 ÷ 10 ÷ 20 Frequency Density 12 10 8 6 4 2 20 30 40 16 14 Time (mins)

Actual raw data plotted against time 10 20 30 40 Actual raw data plotted against time Histogram for data Frequency Density 12 10 8 6 4 2 20 30 40 16 14 Time (mins)

Key Facts About Histograms: A Histogram is the best type of ‘bar chart’ for continuous data A Histogram is a Frequency Chart where the heights of each bar have been divided by the width of the bar The Frequency of each interval in a histogram is therefore the area of the bar not its height. The height is called the Frequency Density.

Q 1) Draw a histogram to represent this data: Time, t, mins Frequency 0 < t ≤ 5 2 5 < t ≤ 15 6 15 < t ≤ 30 12 30 < t ≤ 60

Q 1) Draw a histogram to represent this data: Time, t, mins Frequency F.D. 0 < t ≤ 5 2 0.4 5 < t ≤ 15 6 0.6 15 < t ≤ 30 12 0.8 30 < t ≤ 60 0.8 0.7 Frequency Density 0.6 0.5 0.4 0.3 0.2 0.1 10 20 30 40 50 60 Time (mins)

Summary: (The four main features) The heights of the bars now show frequency density. Frequency density is found by dividing the frequency of the interval by the width of the interval Eg The 1-5 interval had a frequency of 25, so the new height is 25 ÷ 4 = 6.25 The frequency of the interval is now equal to the area of the bar. 12 10 8 6 4 2 20 30 40 16 14 Frequency Density Time (mins) A Histogram is only used for continuous data. It has a continuous scale along the x-axis. The bars are drawn to show the interval widths.

Do the questions: Everyone should do Question 1. I would recommend that most of you do Question 2 (as only 1 of you got it right in the test in July), but if you want a challenge try doing questions 3 and 4 instead which are about applying your knowledge backwards.

Plenary: What are the four main features of a histogram?

Plenary: What are the four main features of a histogram? Continuous scale along the x-axis Bars are matched to the scale on the x-axis to show interval widths (often different widths) The area of a bar tells you the frequency, not the height. The heights of the bars are found by dividing the frequency by the interval width. The vertical axis is labelled frequency density

Plenary: What are the four main features of a histogram? Continuous scale along the x-axis Bars are matched to the scale on the x-axis to show interval widths (often different widths) The area of a bar tells you the frequency, not the height. The heights of the bars are found by dividing the frequency by the interval width. The vertical axis is labelled frequency density What do you think the 3/4 marks for each exam question might be awarded for?

Extension - Making things easier: What happens when we work out the frequency densities for these intervals? 1000 2000 3000 4000