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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