April 13, 2023

Grouping Data

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3. Representing continuous data.

April 13, 2023

Sometimes data is continuous. That means it doesn't fit into distinct categories, such as shoe size, but changes gradually.Height, weight, length, time and temperature are all continuous. There is a continuous scale that can be read at an infinite number of points.

Explanation

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If you record a measurement with a figure, such as 18 cm, it is unlikely that this is actually exact.

April 13, 2023

This credit card card is about 8 cm long…

but if we look closer…

we can see that it is 8.2 cm long…

and if we look closer still…

we see that it is 8.2 cm, and a bit more.

Explanation

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April 13, 2023

Even when you use decimal places, you are unlikely to write an absolutely exact measurement.

So when you make any measurement, you are probably rounding it up or down, whether to a decimal place, or a whole figure.

You are in fact grouping the data at the level of accuracy you choose to use.

When you say something is ‘18 cm’, you may mean…

‘any reading equal to, or more than 17.5 cm and less than 18.5 cm’.

Explanation

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April 13, 2023

‘any reading equal to, or more than 17.5 cm and less than 18.5 cm’.

This can be written 17.5 ≤ w < 18.5

The width of the credit card

is more than, or equal to

17.5

and less than

18.5

To make handling continuous data easier, it can be put into groups larger than a single whole number.

Explanation

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April 13, 2023

These figures show the height in cm of 70 year 10 students at this school.

171 162 169 178 184 174 166

165 171 169 161 178 176 171

158 163 177 165 162 172 168

174 172 174 168 170 175 171

170 172 175 170 176 164 172

180 170 172 180 167 169 172

167 173 155 171 167 174 178

178 173 166 174 170 166 188

173 167 171 161 176 166 180

169 163 174 168 173 161 166

To make this data easier to use, we can group it like this.

Height (cm) Frequency

155 ≤ h < 160 2

160 ≤ h < 165 7

165 ≤ h < 170 19

170 ≤ h < 175 28

175 ≤ h < 180 9

180 ≤ h < 185 4

185 ≤ h < 190 1

You can show continuous data on a bar graph, using proper scales on both axes.

Explanation

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April 13, 2023

You can show continuous data on a bar graph, using proper scales on both axes.

Height (cm) Frequency

155 ≤ h < 160

160 ≤ h < 165

165 ≤ h < 170

170 ≤ h < 175

175 ≤ h < 180

180 ≤ h < 185

185 ≤ h < 190 155 160 165 170 175 180 185 1900

5

10

15

20

25

30

35

Fre

quen

cy

Height (cm)

As the x axis shows a continuous scale, the bars should touch, making this a histogram.

2

7

19

28

9

4

1

Explanation

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April 13, 2023

You can show continuous data on a bar graph, using proper scales on both axes.

Height (cm) Frequency

155 ≤ h < 160

160 ≤ h < 165

165 ≤ h < 170

170 ≤ h < 175

175 ≤ h < 180

180 ≤ h < 185

185 ≤ h < 190 155 160 165 170 175 180 185 1900

5

10

15

20

25

30

35

Fre

quen

cy

Height (cm)

As the x axis shows a continuous scale, the bars should touch, making this a histogram.

2

7

19

28

9

4

1

You can also represent this data using a frequency polygon.

You need to plot the frequency against the mid point of each group.Join the points using straight lines.Do not extend the polygon beyond your plots.

Explanation

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April 13, 2023

You can also represent this data using a frequency polygon.As you do not know the exact value for each item of data, you can only estimate the mean of grouped data.

Do not extend the polygon beyond your plots.

Height (cm) Frequency

155 ≤ h < 160

160 ≤ h < 165

165 ≤ h < 170

170 ≤ h < 175

175 ≤ h < 180

180 ≤ h < 185

185 ≤ h < 190 155 160 165 170 175 180 185 1900

5

10

15

20

25

30

35

Fre

quen

cy

Height (cm)

2

7

19

28

9

4

1

Join the points using straight lines.

Explanation

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April 13, 2023

Add columns for midpoint and midpoint × frequency to your table and a row for totals.As you do not know the exact value for each item of data, you can only estimate the mean of grouped data.

Height (cm) Frequency

155 ≤ h < 160

160 ≤ h < 165

165 ≤ h < 170

170 ≤ h < 175

175 ≤ h < 180

180 ≤ h < 185

185 ≤ h < 190

2

7

19

28

9

4

1

Explanation

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April 13, 2023

Add columns for midpoint and midpoint × frequency to your table and a row for totals.

Height (cm) Frequency

155 ≤ h < 160

160 ≤ h < 165

165 ≤ h < 170

170 ≤ h < 175

175 ≤ h < 180

180 ≤ h < 185

185 ≤ h < 190

2

7

19

28

9

4

1 187.5

182.5

177.5

172.5

167.5

162.5

157.5

187.5

730

11132.5

1597.5

4312.5

3015

975

315

65

Midpoint M × F

Totals

Mean = 11132.5 ÷ 65

The mean = the total of the midpoint × frequency column divided by the total of the frequency column.

= 171.3 cm to 1d.p.

Explanation

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April 13, 2023

The mean = the total of the midpoint × frequency column divided by the total of the frequency column.The range cannot be stated accurately, as we do not know the actual highest and lowest measure.

Height (cm) Frequency

155 ≤ h < 160

160 ≤ h < 165

165 ≤ h < 170

180 ≤ h < 185

185 ≤ h < 190

2

7

19

28

9

4

1 187.5

182.5

177.5

172.5

167.5

162.5

157.5

187.5

730

11132.5

1597.5

4312.5

3015

975

315

65

Midpoint M × F

Totals

Mean = 11132.5 ÷ 65

The modal class is the group with the highest frequency.

= 171.3 cm to 1d.p.

175 ≤ h < 180

170 ≤ h < 175

The range is estimated by taking the mid points of the highest and lowest groups.

Explanation

Range = 157.5 to 187.5EndEndMoreMoreNextNext