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?
Find the population standard deviation of the following sets of scores, correct to two decimal places:
8, \, 20, \, 16, \, 9, \, 9, \, 15, \, 5, \, 17, \, 19, \, 6
- 17, \, 2, \, - 6, \, 9, \, - 17, \, - 9, \, 3, \, 8, \, 5
Given the data in the stem-and-leaf plot, calculate the following correct to two decimal places:
The mean
The population standard deviation
| Leaf | |
|---|---|
| 1 | 0\ 2\ 6 |
| 2 | 6\ 8 |
| 3 | 0 |
| 4 | 5\ 5\ 6 |
| 5 | 6\ 6 |
| 6 | 0 |
| 7 | 7 |
| 8 | 9 |
| 9 | 2 |
Key: 1 \vert 2 = 12
Given the data in the stem-and-leaf plot, calculate the following correct to two decimal places:
The mean.
The population standard deviation.
| Leaf | |
|---|---|
| 1 | 3\ 5\ 5 |
| 2 | 4\ 7 |
| 3 | 8 |
| 4 | 4\ 7\ 8 |
| 5 | 8\ 9 |
| 6 | 6 |
| 7 | 8 |
| 8 | 2 |
| 9 | 8 |
Key: 1 \vert 2 = 12
Find the standard deviation of the following sets of scores, correct to two decimal places, by using the statistics mode on your calculator:
18, \, 7, \, 12, \, 4, \, 16, \, 3, \, 9, \, 16, \, 13, \, 7
- 13, \, 9, \, 7, \, -8, \, 1, \, -13, \, -16, \, 5, \, 10, \, 7
Consider the following column graph:
Find the range of the data set.
Find the mean of the data set.
Find the population standard deviation correct to two decimal places.
The trial times (in seconds) of an athlete for the 100 \text{ m} sprint are recorded below:
12.3, \, 11.8, \, 10.9, \, 11.1, \, 11.2, \, 10.8, \, 11.1, \, 10.8, \, 11.7, \, 12.4
Calculate the population 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.
Meteorologists predicted a huge variation in temperatures throughout the month of April. The temperature each day for the first two weeks of April were recorded as follows:16, \, 18, \, 20.5, \, 21, \, 21, \, 21, \, 21.5, \, 22, \, 22, \, 24, \, 24, \, 25, \, 26, \, 27
State the range of the temperatures.
Calculate the interquartile range of the temperatures.
Find the population standard deviation, correct to one decimal place.
Is there an outlier?
Would the standard deviation or the interquartile range be the best measure of spread to support or counter the prediction of a huge variation in temperatures?
The scores of five diving attempts by a professional diver are recorded below:5.6, \, 6.6, \, 6.3, \, 5.9, \, 6.4
Calculate the mean score, correct to two decimal places.
Calculate the population standard deviation of the scores, correct to two decimal places.
On the sixth dive, the diver scores 8.8.
Is the new score higher or lower than the old mean?
Will the new score increase or decrease the mean?
Will the new score increase or decrease the standard deviation?
If each judge gave the same score for the 6th dive, state the standard deviation of the scores for this dive.
The scores obtained by two classes are given below:
Red Class: 48, \, 51, \, 50, \, 47, \, 51, \, 49, \, 58, \, 57
Blue Class: 44, \, 54, \, 58, \, 61, \, 66, \, 40, \, 51, \,50
Complete the following table. Round all values to two decimal places.
| Mean | Sample standard deviation | |
|---|---|---|
| Red Class | ||
| Blue Class |
Which class performed better? Explain your answer.
Which class produced more consistent results? Explain your answer.
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 (hrs) | Population standard deviation (hrs) | |
|---|---|---|
| Brand X | ||
| Brand Y |
Which brand produces batteries that generally last longer?
Which brand produces batteries that are more consistent?
Consider the following 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 | Population standard deviation | |
|---|---|---|---|
| Set A | |||
| Set B |
Which data set has more variability?
Is range a useful measure to compare variability for these two sets? Explain your answer.
Points scored by two friends over 10 rounds of a game are displayed below:
Quiana: \, 10,\, 31,\, 23,\, 6,\, 5,\, 38,\, 18,\, 19,\, 15,\, 21
Fred: \, 52,\, 51,\, 77,\, 40,\, 61,\, 53,\, 60,\, 81,\, 52,\, 82
Complete the following table. Round all values to one decimal place.
| Mean | Population standard deviation | |
|---|---|---|
| Quiana | ||
| Fred |
Explain what the statistics calculated in part (a) tell us about the two players.
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 (mins) | Population standard deviation (mins) | |
|---|---|---|
| Company X | ||
| Company Y |
Which company generally has better response times?
Which company has more consistent response times?
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.
Compare the mean and standard deviation of the two players.
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 means tell us about the machines?
What does a comparison of the standard deviations tell us about the machines?
| Machine | Mean (g) | Standard deviation (g) |
|---|---|---|
| \text{A} | 52 | 1.5 |
| \text{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 means of the two friends tell us?
What does a comparison of the standard deviations 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 means of the two friends tell us?
What does a comparison of the standard deviations of the two friends tell us?
| Runner | Mean (s) | Standard deviation (s) |
|---|---|---|
| \text{Derek} | 13.1 | 1.2 |
| \text{Sarah} | 14.5 | 0.75 |
Seven millionaires with an average net wealth of \$41 million and a standard deviation of \$7 million are having a party. Suddenly Carlos Slim, who has a net wealth estimated to be \$31 billion, walks into the room.
Find the new average net wealth in the room. Round your answer to the nearest million.
Will the new standard deviation be higher, lower or unchanged from before?
Will the new mode be higher, lower or unchanged from before if at least two of the original seven millionaires have the same net wealth?
Will the range be higher, lower or unchanged from before?
Han, a cricketer, has achieved scores of 52, 20, 68, 70 and 150 in his first five innings this season. In his sixth innings, he scores a duck or 0. Describe how this score in the sixth innings affected the following:
His season batting average.
The standard deviation of his scores.
His median score.
The range of his scores.
A batsman’s mean number of runs is 62 and the standard deviation is 13. In the next match, he makes 50 runs.
Explain how this latest match affects his mean and standard deviation.
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{Mean heart rate} | 90.0 | ||
| \text{Standard deviation of heart rate} | 1.0 | |||
| \text{Tall step} | \text{Mean heart rate} | |||
| \text{Standard deviation of heart rate} |
Which of the combinations of step height and stepping rate generated the highest heart rate?
Which combination of step height and stepping rate showed the least variability?