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

Number of Siblings

Number of Students

0 2

1 5

2 4

3 3

4 1

Distance, d, from school in Miles

Number of Students

0 ≤ t < 1 5

1≤ t < 3 6

3 ≤ t < 5 5

5 ≤ t < 10 4

10 ≤ t < 20 2

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

0 2

1 5

2 4

3 3

4 1

Distance, d, from school in Miles

Number of Students

0 ≤ t < 1 5

1≤ t < 3 6

3 ≤ t < 5 5

5 ≤ t < 10 4

10 ≤ t < 20 2

Draw Bar Charts for both of these sets of data:

0 1 2 3 40

12345

Number of Siblings

Frequency

Students were asked how many brothers or sisters they had

Students were asked how far from school they lived

0 ≤ t< 1 1 ≤ t<3 3 ≤ t<5 5≤ t< 10 10 ≤ t< 20 0

12345

Distance in miles

Frequency 6

0 5 10 15 200

12345

Frequency 6

Number of Siblings

Number of Students

0 2

1 5

2 4

3 3

4 1

Distance, d, from school in Miles

Number of Students

0 ≤ t < 1 5

1≤ t < 3 6

3 ≤ t < 5 5

5 ≤ t < 10 4

10 ≤ t < 20 2

Draw Bar Charts for both of these sets of data:

0 1 2 3 40

12345

Number of Siblings

Frequency

Students were asked how many brothers or sisters they had

Students were asked how far from school they lived

0 5 10 15 200

12345

Frequency 6

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?

0 0 0 0 0 0.5 1.2 1.9 2.7 2.9 3 3.5 4.2 5.6 8.1 8.2 911.7 12.9 14.7 15.5 17.8 23 31

People on a bus route were asked how long they had waited to get on the bus. The times are shown below (in minutes)

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 8.2 911.7 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 8.2 911.7 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 DataDraw 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.

Pupils at a Primary School in Year 6 were asked how long it took them to journey to school. The results are shown below:

Draw a Frequency Polygon for this data.

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

Pupils at a Primary School in Year 6 were asked how long it took them to journey to school. The results are shown below:

Draw a Frequency Polygon for this data.

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

Lesson ObjectiveConsolidate knowledge of Frequency ChartsBe able to draw Frequency Polygons

1) What are the main differences between a Frequency Chart and a Bar Chart?

2) What is a Frequency Polygon?

Key points:Draw a Bar Chart for Discrete DataDraw 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 PolygonsWhat is wrong with these frequency polygons?.

People are really bad at drawing Frequency PolygonsWhat 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 ObjectiveUnderstand the drawbacks of a Frequency Chart for plotting Continuous DataBe 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

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

0

0 10 20 30 40

3025201510 5 0

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

0

0 10 20 30 40

3025201510 5 0

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 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.

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

0

0 10 20 30 40

We like the frequency chart because:

1) It shows the unequal interval widths clearly

2) It shows the frequency

BUT What about the proportions?

Frequency

Time (mins)

30

25

20

15

10

5

0

0 10 20 30 40

Time (mins)

FrequencyDensity

Time (mins)

Frequency 30

25

20

15

10

5

0

0 10 20 30 40

12

10

8

6

4

2

0

0 10 20 30 40

16

14

÷ 4 ÷ 5 ÷ 10 ÷ 20

0 10 20 30 40

Actual raw data plotted against time

FrequencyDensity

Time (mins)

12

10

8

6

4

2

0

0 10 20 30 40

16

14

Histogram for data

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.

Time, t, mins Frequency

0 < t ≤ 5 2

5 < t ≤ 15 6

15 < t ≤ 30 12

30 < t ≤ 60 12

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 12 0.4

0.6

0.5

0.4

0.3

0.2

0.1

0

0 10 20 30 40

0.8

0.7

50 60

Time (mins)

Frequency

Density

Q 1) Draw a histogram to represent this data:

Summary: (The four main features)

FrequencyDensity

Time (mins)

12

10

8

6

4

2

0

0 10 20 30 40

16

14

A Histogram is only used for continuous data. It has a continuous scale along the x-axis.

The frequency of the interval is now equal to the area of the bar.

The bars are drawn to show the interval widths.

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

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?

1. Continuous scale along the x-axis

2. Bars are matched to the scale on the x-axis to show interval widths (often different widths)

3. 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.

4. The vertical axis is labelled frequency density

Plenary:

What are the four main features of a histogram?

What do you think the 3/4 marks for each exam question might be awarded for?

1. Continuous scale along the x-axis

2. Bars are matched to the scale on the x-axis to show interval widths (often different widths)

3. 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.

4. The vertical axis is labelled frequency density

Extension - Making things easier:

What happens when we work out the frequency densities for these intervals?

0 30 40 50 60 70 80 90 100 110

0 1000 2000 3000 4000