10.03 Measuring the centre and spread
Consider the following set of scores: 9, \, 4, \, 14, \, 19, \, 20, \, 15, \, 12
Sort the scores in ascending order.
Find the total number of scores.
Find the median.
Find the mode of the following set of scores:
1, \, 1, \, 1, \, 5, \, 5, \, 9, \, 9, \, 9, \, 9, \, 10, \, 10, \, 10, \, 14, \, 14, \, 14, \, 20
3, \, 3, \, 6, \, 8, \, 8, \, 8, \, 8, \, 10, \, 14, \, 14, \, 14, \, 14, \, 18
Find the median of the following set of scores:
2, \, 5, \, 5, \, 7, \ 9
1, \, 9, \, 2, \, 4, \, 6, \, 7, \, 9
7, \, 8, \, 3, \, 2
3, \, 8, \, 13, \, 17, \, 19, \, 24, \, 26, \, 27
Find the mean of the following scores: \,2,\, 8, \, 17, \, 27, \, 29.
Consider the set of data: 1, \, 2, \, 2, \, 4, \, 4, \, 5, \, 6, \, 6, \, 8, \, 8, \, 8, \, 9, \, 9If one score of 8 is changed to a 9, state the measure(s) of centre that would be altered.
Consider this set of data that represents the number of apps on six people’s phones:11, \, 12, \, 15, \, 17, \, 19, \,19If each person downloads another 7 apps, state the measure(s) of centre that would be altered.
The following five numbers have a mean of 11:
11, 13, 9, 13, 9
If a new number is added that is smaller than 9, describe the effect on the mean.
Consider the following data sets:
Set A: \, 5, \, 2, \, 5, \, 6, \, 6, \, 3
Set B: \, 26, \, 12, \, 14, \, 7, \,16
Which set has the lowest mean?
Which set has the lowest median?
Which data set has the lowest mode?
Set A: \, 87, \, 2, \, 20, \, 20, \, 8, \, 10
Set B: \, 11, \, 8, \, 8, \, 48, \, 2, \, 17
Which data set has the highest median?
Set A: \, 2, \, 8, \, 11, \, 17
Set B: \, 8, \, 20, \, 20, \, 48, \, 87
Determine whether the following statements are true or false:
Two sets of data have the same highest and lowest values. This means they must have the same mode.
Two sets of data that have the same highest and lowest values must have the same range.
If two sets of data have the same median then the data sets must themselves be the same.
If two sets of data have very different modes then the highest values cannot be the same.
State the mode of this data set:
| Score | Frequency |
|---|---|
| 3 | 2 |
| 4 | 4 |
| 5 | 8 |
| 6 | 3 |
| 7 | 5 |
| 8 | 3 |
State the modal class of this data set:
| Class | Frequency |
|---|---|
| 30-39 | 3 |
| 40-49 | 3 |
| 50-59 | 4 |
| 60-69 | 2 |
| 70-79 | 5 |
| 80-89 | 8 |
For the following data set:
Find the median.
Find the mode.
| Score | Frequency | Cumulative frequency |
|---|---|---|
| 3 | 8 | 8 |
| 4 | 2 | 10 |
| 5 | 3 | 13 |
| 6 | 5 | 18 |
| 7 | 3 | 21 |
| 8 | 4 | 25 |
For the following grouped data:
Find the median class.
Find the modal class.
| Class | Frequency |
|---|---|
| 6-10 | 8 |
| 11-15 | 2 |
| 16-20 | 5 |
| 21-25 | 3 |
| 26-30 | 4 |
| 31-35 | 3 |
Consider the data provided in the table:
Calculate the range.
State the mode.
Determine the median.
| Score | Frequency |
|---|---|
| 68 | 16 |
| 69 | 41 |
| 70 | 30 |
| 71 | 31 |
| 72 | 49 |
| 73 | 29 |
Determine the mean for the following data set:
| \text{Score } (x) | \text{Frequency } (f) | xf |
|---|---|---|
| 4 | 8 | 32 |
| 5 | 6 | 30 |
| 6 | 3 | 18 |
| 7 | 8 | 56 |
| 8 | 2 | 16 |
| 9 | 8 | 72 |
Consider the frequency table:
Complete the table using the data set below:
2, \, 7, \, 3, \, 6, \, 3, \, 2, \, 2, \, 2, \, 5, \, 6, \, 3, \, 2, \, 2, \\ 3, \, 5, \, 7, \, 4, \, 2, \, 4, \, 3, \, 2, \, 4, \, 4, \, 6, \, 7Hence find the mean, correct to two decimal places.
Find the median score.
| \text{Score } (x) | \text{Frequency } (f) | xf |
|---|---|---|
| 2 | ||
| 3 | ||
| 4 | ||
| 5 | ||
| 6 | ||
| 7 |
Consider the frequency table:
| \text{Class} | \text{Class centre } (x) | \text{Frequency } (f) | xf |
|---|---|---|---|
| 11-15 | 13 | ||
| 16-20 | 18 | ||
| 21-25 | 23 | ||
| 26-30 | 28 | ||
| 31-35 | 33 | ||
| 36-40 | 38 |
Complete the table using the data set below:
14, \, 36, \, 17, \, 25, \, 15, \, 36, \,19, \, 29, \, 38, \, 23, \, 34, \, 18, \, 34, \\ 31, \, 36, \, 25, \, 32, \, 34, \, 39, \, 26, \, 29, \, 21, \, 37, \, 39, \, 38Hence estimate the mean.
Consider the frequency table:
| \text{Class} | \text{Class centre } (x) | \text{Frequency } (f) | xf |
|---|---|---|---|
| 21-25 | 23 | 7 | |
| 26-30 | 28 | 8 | |
| 31-35 | 33 | 9 | |
| 36-40 | 38 | 6 | |
| 41-45 | 43 | 3 | |
| 46-50 | 48 | 2 |
Complete the frequency table.
Hence estimate the mean, correct to one decimal place.
Calculate the mean for the following data set correct to one decimal place.
| \text{Class} | \text{Class centre } (x) | \text{Frequency } (f) | xf |
|---|---|---|---|
| 11-15 | 13 | 4 | 52 |
| 16-20 | 18 | 3 | 54 |
| 21-25 | 23 | 4 | 92 |
| 26-30 | 28 | 6 | 168 |
| 31-35 | 33 | 8 | 264 |
| 36-40 | 38 | 8 | 304 |
Consider the following table:
Estimate the mean, correct to one decimal place.
State the modal class.
Find the median class.
| Score | Frequency |
|---|---|
| 1-4 | 1 |
| 5-8 | 5 |
| 9-12 | 10 |
| 13-16 | 5 |
| 17-20 | 3 |
Consider the following bar chart:
Find the range.
State the mode.
Determine the mean, correct to two decimal places.
Consider the following dot plot:
Find the total number of scores.
Find the median score.
Find the mode.
Find the range.
State the mode of the data set from the following graphical representations:
Find the median of the data set from the following graphical representations:
Find the mean for the data set from the following graphical representations, rounding your answers to one decimal place:
For each of the given data sets, find the following to two decimal places if necessary:
Mean
Median
Mode
Range
| Leaf | |
|---|---|
| 6 | 2\ 7 |
| 7 | 1\ 2\ 2\ 4\ 7\ 9 |
| 8 | 0\ 1\ 2\ 5\ 7 |
| 9 | 0\ 1 |
Key: 6\vert 2=62
\text{ }
| Leaf | |
|---|---|
| 2 | 4 |
| 3 | 0\ 5\ 5\ 5 |
| 4 | 0\ 2 |
| 5 | 0\ 2\ 9\ 9 |
| 6 | 3\ 3 |
| 7 | 0\ 1 |
| 8 | 0\ 1 |
| 9 | 0\ 0\ 5 |
Key: 2\vert 4=24
A diver measures how long she can hold her breath underwater over several dives. If the median time is 2.1 minutes, this means that:
Most of the time she held her breath for less than 2.1 minutes.
The longest she held her breath is 4.2 minutes.
The shortest time she held her breath is 1.05 minutes.
Most of the time she held her breath for longer than 2.1 minutes.
Half the dives she was able to hold her breath longer than 2.1 minutes.
A real estate agent wanted to determine a typical house price in a certain area. He gathered the selling price of some houses (in dollars):327\,000, \, 376\,000, \, 424\,000, \, 439\,000, \, 444\,000, \, 469\,000, \, 472\,000, \, 475\,000, \, 485\,000, \, 496\,000
Calculate the mean house price.
What percentage of the house prices exceeded the mean?
Determine the median house price.
What percentage of house prices exceeded the median?
Each student in the class was asked to write down the number of siblings they had. The teacher recorded the results in the given dot plot:
How many students are there in the class?
If none of the students share the same siblings, then how many siblings are there in total?
Find the mean number of siblings for a student in this class, correct to one decimal place.
Find the mean number of children in a family for a student in this class, correct to one decimal place.
The given dot plot shows the number of goals scored across each of Rosey's soccer games:
How many games were played in total?
How many goals were scored in total?
Find the mean number of goals per game, correct to one decimal place.
In a study, a group of people were shown 30 names, and after one minute they were asked to recite as many names by memory as possible. The results are presented in the dot plot:
How many people took part in the study?
State the largest number of names someone remembered.
State the smallest number of names someone remembered.
Find the range of the data.
Find the median score.
A cyclist measured his heart rate immediately after finishing each event in which he competed. The results are recorded in the given stem-and-leaf plot:
How many events did the cyclist compete in?
Find his mean post event heart rate.
| Leaf | |
|---|---|
| 16 | 2 |
| 17 | 3\ 8 |
| 18 | 4\ 5\ 6\ 9 |
| 19 | 5\ 5 |
Key: 12|3=123
The scores for a recent history test are shown in the stem-and-leaf plot. The maximum possible score on the test was 100.
How many students took the test?
Find the mean test score for the class.
| Leaf | |
|---|---|
| 6 | 2\ 3 |
| 7 | 2\ 4\ 9 |
| 8 | 3\ 4\ 9\ 9 |
| 9 | 1\ 1\ 5 |
Key: 8|3=83
The size of each earthquake that occurred in a region over a three year period, measured from 0 to 9.9, is recorded in a stem-and-leaf plot:
How many earthquakes in total were recorded?
Find the mean number of earthquakes per year in the region.
It was found that the combined total of all earthquake sizes was 87. Find the mean size of an earthquake that occurred during the period, correct to three decimal places.
| Leaf | |
|---|---|
| 1 | 0\ 0\ 2\ 3\ 5\ 6\ 6\ 7\ 9 |
| 2 | 3\ 8 |
| 3 | 3\ 5\ 7 |
| 4 | 1\ 2\ 2\ 3 |
| 5 | 8\ 9 |
| 6 | 5 |
| 7 | 3\ 6 |
| 8 | 7 |
Key: 5|2=5.2
Find the range of the following set of scores:
20, \, 19, \, 3, \, 19, \, 18, \, 3, \, 16, \, 3
8, \, - 5, \, - 8, \, 4, \, 2, \, 8, \, -9, \, 11
The range of a set of scores is 5, and the highest score is 18. Determine the lowest score in the set.
A group of students had a range in marks of 11 and the lowest score was 5. Determine the highest score in the group.
For each of the following sets of scores:
Sort the scores in ascending order.
Find the number of scores.
Find the median.
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
40, 39, 15, 17, 10, 6, 24
- 4 , - 6 , - 1 , 7, 9, 7, 9
8, 20, 19, 4, 15, 14, 10
42, 28, 22, 40, 20, 54, 32, 43
84, 85, 79, 71, 69, 88, 82, 78
102, 115, 110, 113, 100
228, 205, 198, 202, 207, 197
19, 29, 55, 22, 15, 46, 35, 9, 27, 40
Consider the following data set containing 30 scores:
10, 13, 15, 24, 32, 42, 46, 53, 58, 64
11, 14, 16, 27, 33, 42, 49, 53, 60, 67
11, 15, 18, 28, 37, 44, 51, 55, 61, 67
Find the median.
Find the interquartile range.
Find the interquartile range for the following data set containing 20 scores:
5, 8, 11, 12, 12, 14, 15, 15, 18, 18, 19, 21, 21, 25, 29, 32, 33, 37, 38, 38
For each of the following data sets:
Find the total number of scores.
Find the median.
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
33, 38, 50, 12, 33, 48, 41
- 3 , - 3 , 1, 9, 9, 6, - 9
| Leaf | |
|---|---|
| 2 | 2\ 5\ 6\ 7\ 9 |
| 3 | 0\ 0\ 5\ 6\ 8 |
| 4 | 0\ 0\ 1\ 8\ 9 |
Key: 1 | 2 = 12
| Leaf | |
|---|---|
| 2 | 1\ 3\ 4\ 6\ 9 |
| 3 | 1\ 2\ 2\ 2\ 6 |
| 4 | 2\ 3\ 5\ 6\ 7 |
Key: 1\vert 2 = 12
| Score | Frequency |
|---|---|
| 5 | 3 |
| 13 | 3 |
| 16 | 2 |
| 28 | 2 |
| 31 | 3 |
| 38 | 4 |
| 48 | 2 |
| Score | Frequency |
|---|---|
| 5 | 1 |
| 14 | 1 |
| 18 | 3 |
| 24 | 2 |
| 32 | 1 |
| 38 | 2 |
| 50 | 5 |
Consider the dot plot below:
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
What is the range of this data set?
If there are 78 scores in a set of data, in which position will the lower quartile lie?
The following list shows the number of points scored by a basketball team in each game of their previous season:
59,\, 67,\, 73,\, 82,\, 91,\, 58,\, 79,\, 88,\, 69,\, 84,\, 55,\, 80,\, 98,\, 64,\, 82State the maximum value.
State the minimum value.
Find the median value.
Find the lower quartile score.
Find the upper quartile score.
The following data set shows Ray's scores from her last 13 rounds of golf played:
66, 66, 68, 68, 70, 78, 80, 84, 106, 116, 126, 130, 132
Find her median score.
Find the lower quartile score.
Find the upper quartile score.
Find the interquartile range.
The following data set shows Luke's scores from his last 17 exams:
42, 46, 48, 51, 52, 54, 56, 68, 72, 76, 78, 82, 85, 86, 88, 92, 96
Find his median score.
Find the lower quartile score.
Find the upper quartile score.
Find the interquartile range.
There is a test to measure the Emotional Quotient (EQ) of an individual. Here are the EQ results for 21 people listed in ascending order:
90,\, 90,\, 91,\, 92,\, 93,\, 94,\, 95,\, 95,\, 95,\, 97,\, 99,\, 100,\, 108,\, 114,\, 116,\, 116,\, 117,\, 118,\, 118,\, 122,\, 129
Find the median EQ score.
Find the upper quartile score.
Find the lower quartile score.
Find the interquartile range.
In a competition, a contestant must complete 12 challenges earning as many points as possible. Her scores for the first 11 challenges are:
38, 43, 45, 66, 67, 82, 92, 102, 105, 108, 119
Determine her score in the 12th round if the lower quartile of all of her 12 scores is 55.
The stem-and-leaf plot shows the number of hours students spent studying during an entire semester:
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
| Leaf | |
|---|---|
| 6 | 2\ 7 |
| 7 | 1\ 2\ 2\ 4\ 7\ 9 |
| 8 | 0\ 1\ 2\ 5\ 7 |
| 9 | 0\ 1 |
Key: 5\,\vert\,2\,=\,52
The dot plot shows the ages of customers in a mobile phone store in one day:
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
The dot plot shows the number of sit-ups achieved by 25 students in a physical education exam:
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
Below is the luggage weight of 30 passengers:
Find the mean luggage weight to two decimal places.
Determine the following in kilograms:
Median
Lower Quartile
Upper Quartile
In which quartile does the mean lie?
| Weight | Frequency |
|---|---|
| 16 | 4 |
| 17 | 5 |
| 18 | 5 |
| 19 | 3 |
| 20 | 4 |
| 21 | 6 |
| 22 | 3 |
A group of students were asked how many phone calls they had made the previous day. The information was collected in the following frequency table:
How many students were surveyed?
Find the range of the data.
Find the interquartile range.
| Phone calls | Frequency |
|---|---|
| 0 | 8 |
| 1 | 5 |
| 2 | 10 |
| 3 | 6 |
| 4 | 6 |
| 5 | 7 |
| 6 | 3 |
| 7 | 2 |
A sample of boxes of matches were selected for quality control and the number of matches in each box recorded in the given frequency table:
How many matchboxes were sampled?
Find the range of the data.
Find the interquartile range.
| Matches | Frequency |
|---|---|
| 45 | 1 |
| 46 | 2 |
| 47 | 5 |
| 48 | 3 |
| 49 | 7 |
| 50 | 25 |
| 51 | 6 |
| 52 | 2 |
A group of students were asked how many phone calls they had made the previous day. The information was collected in the given frequency table:
How many students were surveyed?
Find the range of the data.
Find the interquartile range.
| Phone calls | Frequency |
|---|---|
| 0 | 8 |
| 1 | 5 |
| 2 | 10 |
| 3 | 6 |
| 4 | 6 |
| 5 | 7 |
| 6 | 3 |
| 7 | 2 |
Consider the following bar chart:
Organise the data into a frequency table.
Find the median score using the distribution table.
Find the lower quartile score.
Find the upper quartile score.
Find the interquartile range.
The column graph shows the number of pets that each student in a class owns:
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
Dylan records the number of fish he catches for each fishing trip over a period of time:
How many fishing trips did he go on?
Find the range in the number of fish caught.
Find the interquartile range.
The bar graph shows the marks (out of 10) that students received on a spelling test:
Find the lower quartile of the set of scores.
Find the upper quartile of the set of scores.
Find the interquartile range.
Explain the effect the scores less than 7 have on the summary statistics.
The stem-and-leaf plot shows the batting scores of two cricket teams, A and B:
What is the median score of Team A?
What is the median score of Team B?
What is the range of Team A’s scores?
What is the range of Team B’s scores?
What is the interquartile range of Team A’s scores?
What is the interquartile range of Team B’s scores?
Explain the differences between Team A and Team B's distribution of scores.
| Team A | Team B | |
|---|---|---|
| 6\ 5\ 4 | 6 | 0\ 5\ 8 |
| 8\ 5\ 4\ 0 | 7 | 1\ 5\ 7 |
| 8\ 6 | 8 | 2\ 3\ 7\ 9 |
| 9 | 2\ 5 |
Key: 6\,\vert \,1\,\vert\, 2 =\, 12 \,and\,16
To gain a place in the main race of a car rally, teams must compete in a qualifying round. The median time in the qualifying round determines the cut off time to make it through to the main race. Below are some results from the qualifying round:
75\% of teams finished in 159 minutes or less.
25\% of teams finished in 132 minutes or less.
25\% of teams finished between with a time between 132 and 142 minutes.
Find the cut off time required in the qualifying round to make it through to the main race.
Find the interquartile range in the qualifying round.
In the qualifying round, the ground was wet, while in the main race, the ground was dry. To make the times more comparable, the finishing time of each team from the qualifying round is reduced by 5 minutes.
Find the new median time from the qualifying round.