For each of the given sets of data find the following correct to two decimal places:
8, 20, 16, 9, 9, 15, 5, 17, 19, 6
- 17 , 2, - 6 , 9, - 17 , - 9 , 3, 8, 5
| Leaf | |
|---|---|
| 1 | 0\ 2\ 6 |
| 2 | 6\ 8 |
| 3 | 0 |
| 4 | 5\ 5\ 6 |
| 5 | 6\ 6 |
| 6 | 0 |
| 7 | 7 |
Key: 1 \vert 2 = 12
| Score | Frequency |
|---|---|
| 15 | 13 |
| 16 | 9 |
| 17 | 23 |
| 18 | 19 |
| 19 | 8 |
| 20 | 13 |
| Score | Frequency |
|---|---|
| 68 | 16 |
| 69 | 41 |
| 70 | 30 |
| 71 | 31 |
| 72 | 49 |
| 73 | 29 |
Consider the given column graph:
Find the range of the data set.
Find the mean of the data set, correct to two decimal places.
Find the population standard deviation, correct to two decimal places.
The table shows the number of goals scored by a football team in each game of the year.
In how many games were 0 goals scored?
Find the median number of goals scored.
Find the mean number of goals scored each game.
Find the population standard deviation, correct to two decimal places.
| Goals | Frequency |
|---|---|
| 0 | 3 |
| 1 | 1 |
| 2 | 5 |
| 3 | 1 |
| 4 | 5 |
| 5 | 5 |
Consider the given frequency table:
Complete the table.
Use the class centres to estimate the mean. Round your answer to two decimal places.
Use the class centres to find an estimate for the population standard deviation. Round your answer to two decimal places.
If we used the original ungrouped data to calculate standard deviation, do you expect that the ungrouped data would have a higher or lower standard deviation?
| Class | Class centre | Frequency |
|---|---|---|
| 1-9 | 8 | |
| 10-18 | 6 | |
| 19-27 | 4 | |
| 28-36 | 6 | |
| 37-45 | 8 | |
| \text{Total} |
Use the class centres to find an estimate for the population standard deviation of the data represented in the following grouped frequency table. Round your answer to two decimal places.
| Class | Class centre | Frequency |
|---|---|---|
| 40\leq x \lt 45 | 42.5 | 4 |
| 45 \leq x \lt 50 | 47.5 | 11 |
| 50 \leq x \lt 55 | 52.5 | 16 |
| 55 \leq x \lt 60 | 57.5 | 17 |
| 60 \leq x \lt 65 | 62.5 | 7 |
| 65 \leq x \lt 70 | 67.5 | 12 |
| 70 \leq x \lt 75 | 72.5 | 11 |
| 75 \leq x \lt 80 | 77.5 | 5 |
Use the class centres to find an estimate for the population standard deviation of the data represented in the following histogram. Round your answer to two decimal places.
The scores of five diving attempts by a professional diver are recorded below:
6.4, \, 5.9,\, 5.1,\, 5.1,\, 5.5
Calculate the sample standard deviation of his scores. Round your answer to two decimal places.
On his sixth attempt, the diver scores 9.3. State whether this score would increase or decrease the following:
The mean
The standard deviation
If each judge gave the diver the same score, what would be the sample standard deviation of the judges’ scores?
The trial times (in seconds) of an athlete for the 100 \text{ m} sprint are recorded below:
11.0,\, 10.5,\, 12.2,\, 11.2,\, 11.6,\, 11.7,\, 10.8,\, 12.1,\, 11.0,\, 10.9
Calculate the sample standard deviation of her times. Round your answer to two decimal places.
On her next attempt, she manages to run 100 \text{ m} in 10.1 \text{ s}. Explain what impact this latest attempt would have on her mean and standard deviation.
A batsman's mean number of runs is 62 and the standard deviation is 13. In the next match he makes 50 runs. Explain what impact this latest match would have on his mean and standard deviation.
Meteorologists predicted huge variation in temperatures throughout the month of April. The temperature each day for the first two weeks of April was recorded:
16,\, 16.5,\, 16.5,\, 17.5,\, 19.5,\, 22.5,\, 23.5,\, 23.5,\, 26,\, 26,\, 26.5,\, 27,\, 27.5,\, 28
What is the range of the temperatures?
What is the interquartile range of the temperatures?
What is the sample standard deviation? Round your answer to one decimal place.
Would the sample standard deviation or the interquartile range be the best measure of spread to support or counter a prediction? Explain your answer.
The mean income of people in Country A is \$19\,069. This is the same as the mean income of people in Country B. The standard deviation of Country A is greater than the standard deviation of Country B. In which country is there likely to be the greatest difference between the incomes of the rich and poor?
Two machines A and B are producing chocolate bars with the following mean and standard deviation for the weight of the bars:
What does a comparison of the mean of the two machines tell us?
What does a comparison of the standard deviation of the two machines tell us?
| Machine | Mean (g) | Standard deviation (g) |
|---|---|---|
| A | 52 | 1.5 |
| B | 56 | 0.65 |
Two friends compete in triple jump and the distance of 20 jumps were recorded. The mean and standard deviation for the jumps are shown below:
What does a comparison of the mean of the two friends tell us?
What does a comparison of the standard deviation of the two friends tell us?
| Jumper | Mean (m) | Standard deviation (m) |
|---|---|---|
| \text{William} | 12.6 | 0.8 |
| \text{Kathleen} | 11.6 | 0.4 |
Two friends compete in 100 \text{ m} sprints and the time to complete 50 sprints were recorded. The mean and standard deviation for the sprints are shown below:
What does a comparison of the mean of the two friends tell us?
What does a comparison of the standard deviation of the two friends tell us?
| Runner | Mean (s) | Standard deviation (s) |
|---|---|---|
| \text{Derek} | 13.1 | 1.2 |
| \text{Sarah} | 14.5 | 0.75 |
Two cricketers compare the mean and standard deviation of their runs made per match. They conclude that Tobias is a more consistent batter but Lucy generally scores more runs per match.
What can we say about the comparison of their means and standard deviations?
The table shows the heart rate data of a group of people after exercise:
| Height of step | Stepping rate | Heart rate |
|---|---|---|
| \text{Short step} | \text{Slow} | 89 |
| \text{Short step} | \text{Slow} | 91 |
| \text{Short step} | \text{Medium} | 106 |
| \text{Short step} | \text{Medium} | 105 |
| \text{Short step} | \text{Fast} | 124 |
| \text{Short step} | \text{Fast} | 128 |
| \text{Tall step} | \text{Slow} | 100 |
| \text{Tall step} | \text{Slow} | 96 |
| \text{Tall step} | \text{Medium} | 125 |
| \text{Tall step} | \text{Medium} | 129 |
| \text{Tall step} | \text{Fast} | 132 |
| \text{Tall step} | \text{Fast} | 127 |
Complete the following table. Round all values to one decimal place.
| Height of step | Data | Slow | Medium | Fast |
|---|---|---|---|---|
| \text{Short step} | \text{Average heart rate} | 90.0 | ||
| \text{Standard deviation of heart rate} | 1.0 | |||
| \text{Tall step} | \text{Average heart rate} | |||
| \text{Standard deviation of heart rate} |
Which of the combinations of step height and stepping rate generated the higher heart rate?
Which combination of step height and stepping rate showed the least variability?
The scores obtained by two classes are given below:
-
Red class: \,48,\, 56,\, 49,\, 43,\, 41,\, 44,\, 49,\, 46
Blue class: \,64,\, 52,\, 43,\, 69,\, 66,\, 65,\, 58,\, 42
Complete the following table. Round all values to two decimal places.
| Mean | Sample standard deviation | |
|---|---|---|
| Red class | ||
| Blue class |
Which class performed better?
Which class produced more consistent results?
Consider the following pair of data sets below.
-
Set A: 2,\, 4,\, 5,\, 8,\, 8,\, 9,\, 9,\, 9,\, 10,\, 10,\, 10,\, 11,\, 12,\, 15
Set B: 2,\, 2,\, 3,\, 4,\, 5,\, 5,\, 6,\, 7,\, 9,\, 11,\, 13,\, 13,\, 14,\, 15
Complete the following table. Round values to one decimal place where necessary.
| Range | Interquartile range | Sample standard deviation | |
|---|---|---|---|
| Set A | |||
| Set B |
Which data set has more variability?
Is range a useful measure to compare variability for these two sets?
The life of two brands of batteries are tested using a sample of 10 batteries from each brand. Their battery lives (in hours) are shown below.
-
Brand X: 23.3,\, 19.7,\, 20.7,\, 25.3,\, 22.5,\, 19.1,\, 20.0,\, 20.7,\, 20.7,\, 20.9
Brand Y: 23.2,\, 27.5,\, 25.0,\, 24.5,\, 22.7,\, 29.8,\, 22.9,\, 26.0,\, 26.4,\, 22.6
Complete the following table. Round all values to one decimal place.
| Mean (h) | Sample standard deviation (h) | |
|---|---|---|
| Brand X | ||
| Brand Y |
Which brand produces batteries that generally last longer?
Which brand produces batteries that are more consistent?
Two companies record the wait time for calls to their customer hotlines over 10 calls. The recorded values are given below in minutes.
Company X: 3.1,\, 2.1,\, 3.1,\, 3.2,\, 3.0,\, 2.6,\, 3.8,\, 2.7,\, 2.5,\, 3.5
-
Company Y: 2.2,\, 3.1,\, 3.0,\, 2.5,\, 3.0,\, 3.2,\, 3.2,\, 2.7,\, 2.5,\, 2.5
Complete the following table. Round all values to two decimal places.
| Mean (m) | Sample standard deviation (m) | |
|---|---|---|
| Company X | ||
| Company Y |
Which company generally has better response times?
Which company has more consistent response times?
Points scored by two friends over 10 rounds of a game are displayed below.
Pauline: 10,\, 31,\, 23,\, 6,\, 5,\, 38,\, 18,\, 19,\, 15,\, 21
-
Dave: 52,\, 51,\, 77,\, 40,\, 61,\, 53,\, 60,\, 81,\, 52,\, 82
Complete the following table. Round all values to one decimal place.
| Mean | Sample standard deviation | |
|---|---|---|
| Pauline | ||
| Dave |
Explain what the statistics calculated above tell us about the two players.