Cumulative-frequency graphs and tables - Free Printable

Problem Analysis and Solution

The image contains three cumulative frequency graphs, each representing different datasets. We will solve the problems step by step for each graph.

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#### Graph A: Posted Parcel Weights (Grouped Data)

1. How many packages were weighed?

- The cumulative frequency graph shows the total number of packages on the y-axis.
- The maximum value on the y-axis represents the total number of packages.
- From the graph, the maximum cumulative frequency is 40.
- Therefore, the total number of packages weighed is 40.

2. "The range of weights is 5 kg." Do you agree?

- The range of a dataset is the difference between the maximum and minimum values.
- From the x-axis of the graph:
- The minimum weight is approximately 1 kg.
- The maximum weight is approximately 6 kg.
- The range is calculated as:
\[
\text{Range} = \text{Maximum weight} - \text{Minimum weight} = 6 - 1 = 5 \, \text{kg}
\]
- Therefore, the statement "The range of weights is 5 kg" is correct.

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#### Graph B: Tree Heights in Kelty Wood

1. Estimate:
- The median
- The interquartile range (IQR)

- Median:
- The median is the value where half of the data lies below it and half lies above it.
- On a cumulative frequency graph, the median corresponds to the point where the cumulative frequency is half of the total frequency.
- From Graph B, the total cumulative frequency is 80.
- Half of 80 is 40.
- Locate the point on the graph where the cumulative frequency is 40 and read the corresponding height on the x-axis.
- From the graph, when the cumulative frequency is 40, the height is approximately 5.2 meters.
- Therefore, the median height is 5.2 meters.

- Interquartile Range (IQR):
- The IQR is the range between the first quartile (Q1) and the third quartile (Q3).
- Q1 is the value where 25% of the data lies below it, and Q3 is the value where 75% of the data lies below it.
- For a total cumulative frequency of 80:
- Q1 corresponds to a cumulative frequency of \( \frac{1}{4} \times 80 = 20 \).
- Q3 corresponds to a cumulative frequency of \( \frac{3}{4} \times 80 = 60 \).
- From the graph:
- When the cumulative frequency is 20, the height is approximately 4.0 meters (Q1).
- When the cumulative frequency is 60, the height is approximately 6.2 meters (Q3).
- The IQR is calculated as:
\[
\text{IQR} = Q3 - Q1 = 6.2 - 4.0 = 2.2 \, \text{meters}
\]
- Therefore, the interquartile range is 2.2 meters.

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#### Graph C: Sprint Practice Times

1. Estimate the percentage of runners that were slower than 1 minute 10 seconds.

- First, convert 1 minute 10 seconds into seconds:
\[
1 \, \text{minute} \, 10 \, \text{seconds} = 60 + 10 = 70 \, \text{seconds}
\]
- On the graph, locate the point where the time is 70 seconds on the x-axis and read the corresponding cumulative frequency.
- From the graph, when the time is 70 seconds, the cumulative frequency is approximately 120.
- The total cumulative frequency (maximum value on the y-axis) is 150.
- The percentage of runners who were slower than 70 seconds is calculated as:
\[
\text{Percentage} = \left( \frac{\text{Cumulative frequency at 70 seconds}}{\text{Total cumulative frequency}} \right) \times 100 = \left( \frac{120}{150} \right) \times 100 = 80\%
\]
- Therefore, the percentage of runners who were slower than 1 minute 10 seconds is 80%.

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Final Answers

1. Graph A:
- Total packages weighed: 40
- Agreement with the range: Yes

2. Graph B:
- Median: 5.2 meters
- Interquartile range: 2.2 meters

3. Graph C:
- Percentage of runners slower than 1 minute 10 seconds: 80%

\boxed{40, \text{Yes}, 5.2, 2.2, 80}