18.10 Cumulative frequency tables and graphs (Extended)
Consider the following table:
Find the total number of scores recorded.
Find the number of times a score of 14 occured.
Find the number of times a score less than 13 occured.
| \text{Score } (x) | \text{Cumulative frequency } (cf) |
|---|---|
| 10 | 7 |
| 11 | 15 |
| 12 | 18 |
| 13 | 20 |
| 14 | 26 |
Complete the following table:
| \text{Score } (x) | \text{Frequency } (f) | \text{Cumulative frequency } (cf) |
|---|---|---|
| 18 | 10 | |
| 19 | 9 | |
| 20 | 3 | 22 |
| 21 | ||
| \text{Total:} | 29 |
The number of sightings of the Northern Lights were recorded across various Canadian locations over a period of 1 month. The numbers below represent the number of sightings at each location:
12,\, 8,\, 9,\, 8,\, 11,\, 7,\, 7,\, 11,\, 10,\, 9,\, 9,\, 11,\, 7,\, 10,\, 11,\, 7,\, 8,\, 9,\, 11,\, 9
Construct a cumulative frequency table for this data.
Find the number of locations where there were at least 11 sightings.
Find the number of locations where there were less than 11 sightings.
Find the median number of sightings across all 20 locations.
A pair of dice are rolled 50 times and the numbers appearing on the uppermost face are added to give a score. The results are recorded in the given table:
State the lowest possible score when a single pair of dice are rolled.
State the highest possible score when a single pair of dice are rolled.
Complete the table by finding the cumulative frequency values.
Find the number of times a score of 8 occured.
Find the number of times a score more than 9 occured.
Find the number of times a score of at most 6 occured.
| \text{Score} \\ (x) | \text{Frequency} \\ (f) | \text{Cumulative} \\ \text{frequency } (cf) |
|---|---|---|
| 2 | 1 | |
| 3 | 2 | |
| 4 | 5 | |
| 5 | 5 | |
| 6 | 5 | |
| 7 | 9 | |
| 8 | 7 | |
| 9 | 5 | |
| 10 | 8 | |
| 11 | 1 | |
| 12 | 2 |
Consider the frequency table showing the number of 'holes in one' across golf tournaments:
Find the total number of 'holes in one' across all the tournaments.
In how many tournaments were at most 3 'holes in one' scored?
| Number of 'holes in one' | Tournaments |
|---|---|
| 2 | 5 |
| 3 | 1 |
| 4 | 3 |
| 5 | 4 |
| 6 | 0 |
Families were asked how many times they got the flu during winter. The information has been partially filled out in the following table:
Complete the frequency and cumulative frequency values in the given table.
Find the total number of families that responded.
Find the median number of times a family got the flu.
Find the mean number of times a family got the flu. Round your answer to two decimal places.
Find the number of families who got the flu more than twice.
| \text{Score } \\ (x) | \text{Frequency } \\ (f) | fx | \text{Cumulative} \\ \text{frequency} |
|---|---|---|---|
| 0 | 0 | 0 | |
| 1 | 2 | ||
| 2 | 6 | ||
| 3 | 6 | ||
| 4 | 8 |
Consider the following cumulative frequency curve:
Estimate:
The 60th percentile.
The 10th percentile.
The lower quartile.
The median
The upper quartile.
The interquartile range.
For each of the following graphs use the cumulative frequency curve to estimate:
The median score.
The lower quartile.
The upper quartile.
Consider the following frequency table:
Complete the table by finding the cumulative frequency values.
Construct a cumulative frequency curve for the data.
Find the median.
Use your graph to estimate the 20th percentile.
| \text{Score } (x) | \text{Frequency } (f) | \text{Cumulative}\\ \text{frequency }(cf) |
|---|---|---|
| 134 | 3 | |
| 135 | 2 | |
| 136 | 4 | |
| 137 | 6 | |
| 138 | 4 | |
| \text{Total:} | 19 |
Consider the frequency distribution table below:
| \text{Score } (x) | \text{Frequency } (f) | \text{Cumulative frequency } (cf) |
|---|---|---|
| 1 - 4 | 4 | |
| 5 - 8 | 7 | |
| 9 - 12 | 11 | |
| 13 - 16 | 7 | |
| 17 - 20 | 4 |
Complete the table.
Calculate the total frequency.
Identify the class size.
Describe the shape of the distribution.
Approximately one third of the scores recorded are greater than what score?
For each of the frequency distribution tables below:
Complete the table.
State the modal class.
In which class interval does the median lie?
Using the class centres, estimate the mean to two decimal places.
| Class | Class centre | Frequency | Cumulative frequency |
|---|---|---|---|
| 1-7 | 11 | ||
| 8-14 | 14 | ||
| 15-21 | 10 | ||
| 22-28 | 15 | ||
| 29-35 | 24 | ||
| \text{Total:} |
| Class | Class centre | Frequency | Cumulative frequency |
|---|---|---|---|
| 1-9 | 8 | ||
| 10-18 | 16 | ||
| 19-27 | 4 | ||
| 28-36 | 21 | ||
| 37-45 | 16 | ||
| \text{Total:} |
Consider the frequency distribution table below:
| \text{Score } (x) | \text{Frequency } (f) | \text{Cumulative frequency } (cf) |
|---|---|---|
| 1 - 5 | 7 | |
| 6 - 10 | 15 | |
| 11 - 15 | 8 | |
| 16 - 20 | 11 | |
| 21 - 25 | 2 | |
| 26 - 30 | 1 |
Complete the table.
Calculate the total frequency.
Identify the class size.
Describe the shape of the distribution.
Approximately half of the scores recorded are greater than what score?
Consider the frequency distribution table below:
| \text{Score } (x) | \text{Frequency } (f) | \text{Cumulative frequency } (cf) |
|---|---|---|
| 20 - 24 | 9 | |
| 25 - 29 | 18 | |
| 30 - 34 | 37 | |
| 35 - 39 | 14 | |
| 40 - 44 | 9 | |
| 45 - 49 | 6 | |
| 50 - 54 | 3 |
Complete the table.
Calculate the total frequency.
Identify the class size.
Describe the shape of the distribution.
Approximately one third of the scores recorded are greater than what score?
Consider the set of scores given below:
58,\, 59,\, 70,\, 64,\, 69,\, 73,\, 64,\, 68,\, 59,\, 60,\, 54,\, 73
Complete the following frequency table:
| \text{Class interval} | \text{Class centre } (cc) | \text{Frequency } (f) | f \times cc | \text{Cumulative} \\ \text{frequency} |
|---|---|---|---|---|
| 51-55 | ||||
| 56-60 | ||||
| 61-65 | ||||
| 66-70 | ||||
| 71-75 | ||||
| \text{Total} |
State the modal class.
Using the class centres, estimate the mean to one decimal place.
Scientists wanting to determine the effect of fatigue on reaction time while driving. They measured the reaction time, t, of several drivers at night. The results are presented in the following table:
| Reaction time (seconds) | Class centre | Frequency | Cumulative frequency |
|---|---|---|---|
| 0.01 \leq t \lt 0.05 | 0.03 | 29 | 29 |
| 0.05 \leq t \lt 0.09 | 0.07 | 32 | 61 |
| 0.09 \leq t \lt 0.13 | 0.11 | 39 | 100 |
| 0.13 \leq t \lt 0.17 | 0.15 | 31 | 131 |
| 0.17 \leq t \lt 0.21 | 0.19 | 36 | 167 |
| 0.21 \leq t \lt 0.25 | 0.23 | 33 | 200 |
Using the class centres, calculate an estimate of:
One researcher wants to reduce the amount of data by increasing the size of each class interval. Complete the table:
| Reaction Time (seconds) | Class Centre | Frequency | Cumulative frequency |
|---|---|---|---|
| 0.01 \leq t \lt 0.09 | |||
| 0.09 \leq t \lt 0.17 | |||
| 0.17 \leq t \lt 0.25 |
Using the new class centres, calculate an estimate of:
By increasing the size of each class interval, by what percentage has the mean changed?