Averages And Range

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Lesson notes

Discrete & Continuous Data

Continuous data can take any numerical value on a scale (e.g., height, weight, time).
Discrete data can only take particular values, often integers (e.g., number of people, shoe sizes).
If you need a scale to measure it, it is likely continuous.

Mean, Median & Mode

Mode: value that appears most often; can be more than one mode.
Median: middle value when data is ordered; for even number of values, take midpoint of the two middle values.
Mean: sum of values divided by number of values; can be a decimal.
Use median when data has extreme values (outliers); avoid mode if more than one mode; mode is the only average for non-numerical data.

Calculations with the Mean

Mean = total of values ÷ number of values.
Rearrange: total of values = mean × number of values.
Use totals before and after to find missing values or effect of adding/removing a value.

Averages from Tables

Mode: data value with highest frequency.
Median: the ((n+1)/2)th value; use cumulative frequency to locate it.
Mean: total of (data value × frequency) divided by total frequency.
Range: largest data value minus smallest data value (not difference of frequencies).

Number of pets per house

Averages from Grouped Data

Grouped data uses class intervals; exact mean cannot be found, only estimated.
Estimated mean: use midpoint of each interval as representative value, then total of (midpoint × frequency) divided by total frequency.
Modal class: class interval with highest frequency.
Median class: class interval containing the ((n+1)/2)th value.

Range & Interquartile Range

Range = highest value – lowest value; measures spread but affected by outliers.
Lower quartile (LQ) is median of lower half of data; upper quartile (UQ) is median of upper half.
Interquartile range (IQR) = UQ – LQ; measures spread of middle 50% and is not affected by outliers.

Comparing Data Sets

Compare averages (mode, median, mean) and spreads (range, IQR).
Write conclusions: state which is higher/lower and what that means in context (e.g., 'on average, slugs travel further').
Use phrases like 'more consistent', 'less variation' for smaller spread.
Be aware of limitations: small sample size, bias, or influence of outliers.

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Practice questions

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1.State whether the following data is discrete or continuous: The weights of dogs participating in a dog show.
Easy
- ADiscrete
- BContinuous
- CNeither
- DBoth
2.Find the range of the shoe sizes: 6, 7, 8, 9, 10, 11, 12.
Easy
- A6
- B7
- C5
- D12
3.For the data set: 1, 2, 2, 5, 5, 6, what is the mode?
Easy
- A2
- B5
- C2 and 5
- DNo mode
4.Find the median of the numbers: 4, 2, 3.
Easy
- A3
- B2
- C4
- D2.5
5.Calculate the mean of: 1, 2, 6.
Easy
- A3
- B4
- C2
- D3.5
6.Roberto records the value of each of his coins. The table shows the value and frequency. | Value (cents) | 1 | 2 | 5 | 10 | 20 | 50 | |---|---|---|---|---|---|---| | Frequency | 3 | 1 | 3 | 2 | 4 | 2 | Find the range.
Medium
- A49 cents
- B50 cents
- C51 cents
- D1 cent
7.For the same coin data, find the median.
Medium
- A5 cents
- B10 cents
- C20 cents
- D2 cents
8.A teacher records the number of days each of 24 students are absent. Frequency table: Days: 0,1,2,3,4,5; Frequency: 10,8,3,2,0,1. Find the mode.
Easy
- A0 days
- B1 day
- C2 days
- D5 days

More questions

9.For the same absence data, find the median.
Medium
- A0 days
- B1 day
- C2 days
- D0.5 days
10.40 people were asked how many times they visited the cinema. Table: Visits:0,1,2,3,4,5,6,7; Frequency:5,5,6,6,7,3,6,2. Find the mode.
Medium
- A4
- B5
- C6
- D3
11.State whether the following data is discrete or continuous: The number of books in a library.
Easy
- ADiscrete
- BContinuous
- CBoth
- DNeither
12.Find the median of the numbers: 7, 3, 9, 5, 1.
Easy
- A5
- B3
- C7
- D9
13.The heights (in cm) of 5 plants are: 12, 15, 14, 10, 19. Calculate the mean height.
Easy
- A14 cm
- B13 cm
- C15 cm
- D14.5 cm
14.The table shows the number of goals scored by a football team in 10 matches. | Goals | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | Frequency | 2 | 3 | 2 | 2 | 1 | Find the mode.
Easy
- A1 goal
- B2 goals
- C0 goals
- D3 goals
15.For the same goal data (goals: 0,1,2,3,4; frequencies: 2,3,2,2,1), find the median number of goals.
Medium
- A1
- B2
- C1.5
- D2.5
16.A class of 24 students has a mean height of 1.56 m. A new student joins, and the mean becomes 1.57 m. Find the height of the new student.
Medium
- A1.81 m
- B1.75 m
- C1.68 m
- D1.62 m
17.The time taken for each of 120 students to complete a puzzle is shown: Time (t minutes): 20<t≤25, 25<t≤30, 30<t≤35, 35<t≤40, 40<t≤45; Frequency: 44,32,28,12,4. Write down the modal time interval.
Easy
- A20 < t ≤ 25
- B25 < t ≤ 30
- C30 < t ≤ 35
- D35 < t ≤ 40
18.For the same puzzle time data (20<t≤25:44, 25<t≤30:32, 30<t≤35:28, 35<t≤40:12, 40<t≤45:4), write down the interval containing the median time.
Medium
- A20 < t ≤ 25
- B25 < t ≤ 30
- C30 < t ≤ 35
- D35 < t ≤ 40
19.Find the range of the shoe sizes: 6, 6.5, 7, 7.5, 8, 9, 10, 11, 12.
Easy
- A6
- B5
- C7
- D6.5
20.State whether the following data is discrete or continuous: The time taken to swim 100 metres by each member of a swimming club.
Easy
- AContinuous
- BDiscrete
- CBoth
- DNeither
21.The weights (kg) of 10 puppies are: 2.3, 2.5, 2.8, 3.1, 3.4, 3.6, 3.9, 4.2, 4.5, 5.0. Find the interquartile range.
Medium
- A1.7 kg
- B1.4 kg
- C2.0 kg
- D1.9 kg
22.For the same test scores (21,21,23,26,25,21,22,20,21,23,23,27,24,21), find the median.
Medium
- A22
- B21
- C23
- D22.5
23.The weights of 20 Labrador puppies are recorded in the table. | Weight (w kg) | Frequency | |---|---| | 3 ≤ w < 3.5 | 2 | | 3.5 ≤ w < 4 | 4 | | 4 ≤ w < 4.5 | 6 | | 4.5 ≤ w < 5 | 5 | | 5 ≤ w < 5.5 | 2 | | 5.5 ≤ w < 6 | 1 | Estimate the mean weight of the puppies.
Medium
- A4.35 kg
- B4.50 kg
- C4.25 kg
- D4.00 kg
24.The table shows the time (t minutes) for 120 people in a supermarket. | Time (T min) | Frequency | |---|---| | 10 < T ≤ 30 | 16 | | 30 < T ≤ 40 | 18 | | 40 < T ≤ 45 | 22 | | 45 < T ≤ 50 | 40 | | 50 < T ≤ 60 | 21 | | 60 < T ≤ 70 | 3 | Use midpoints to estimate the mean time.
Hard
- A42.8 min
- B44.3 min
- C40.5 min
- D45.0 min
25.The table shows the amount of money, $x, given to a charity by each of 60 people. | Amount ($x) | Frequency | |---|---| | 0 < x ≤ 20 | 21 | | 20 < x ≤ 25 | 16 | | 25 < x ≤ 35 | 6 | | 35 < x ≤ 50 | 10 | | 50 < x ≤ 100 | 7 | Calculate an estimate of the mean.
Hard
- A$28.33
- B$32.50
- C$25.00
- D$30.20
26.The heights (h metres) of 120 boys are recorded. | Height (h m) | Frequency | |---|---| | 1.3 < h ≤ 1.4 | 7 | | 1.4 < h ≤ 1.5 | 18 | | 1.5 < h ≤ 1.6 | 30 | | 1.6 < h ≤ 1.7 | 24 | | 1.7 < h ≤ 1.8 | 27 | | 1.8 < h ≤ 1.9 | 14 | Calculate an estimate of the mean height.
Hard
- A1.62 m
- B1.58 m
- C1.65 m
- D1.60 m
27.The test scores of 14 students are: 21, 21, 23, 26, 25, 21, 22, 20, 21, 23, 23, 27, 24, 21. Find the mean.
Easy
- A22.7
- B23.0
- C22.0
- D21.5
28.The table shows the marks scored by 40 students in a test. | Mark | Frequency | |---|---| | 5 | 8 | | 6 | 5 | | 7 | 11 | | 8 | 7 | | 9 | 5 | | 10 | 4 | Calculate the mean mark.
Medium
- A7.0
- B7.2
- C6.8
- D7.5
29.The speed, v km/h, of each of 200 cars is measured. | Speed (v km/h) | Frequency | |---|---| | 0 < v ≤ 20 | 16 | | 20 < v ≤ 40 | 34 | | 40 < v ≤ 45 | 62 | | 45 < v ≤ 50 | 58 | | 50 < v ≤ 60 | 26 | | 60 < v ≤ 80 | 4 | Calculate an estimate of the mean speed.
Hard
- A41.4 km/h
- B42.5 km/h
- C40.0 km/h
- D45.0 km/h
30.The time taken for each of 120 students to complete a puzzle is shown. | Time (t min) | Frequency | |---|---| | 20 < t ≤ 25 | 44 | | 25 < t ≤ 30 | 32 | | 30 < t ≤ 35 | 28 | | 35 < t ≤ 40 | 12 | | 40 < t ≤ 45 | 4 | Calculate an estimate of the mean time.
Medium
- A28.3 min
- B30.0 min
- C27.5 min
- D29.2 min
31.The weight of a dog is an example of discrete data.
Easy
True or false?
32.What is the mode of the data set: 1, 2, 2, 5, 5, 6?
Easy
- A2
- B5
- C2 and 5
- D6
33.Match each type of data with the correct example.
Easy
- Discrete data
- Continuous data
- The number of books in a library
- The weight of a dog
34.Arrange the following steps in order to find the interquartile range of a data set.
Medium
- Find the median of the lower half
- Find the median of the upper half
- Order the data
- Subtract lower quartile from upper quartile
35.Which of the following are measures of central tendency? (Select all that apply.)
Easy
- AMean
- BRange
- CMedian
- DMode
- EInterquartile range
36.Which of the following statements about the median are true? (Select all that apply.)
Medium
- AIt is the middle value when data is ordered
- BIt is affected by extreme values
- CIt can be found for grouped data only as an interval
- DFor an even number of values, it is the mean of the two middle values
37.The range of a data set is the difference between the highest and lowest values.
Easy
True or false?
38.Which of the following is an example of continuous data?
Easy
- ANumber of students in a class
- BShoe size of a person
- CWeight of a bag of flour
- DNumber of cars in a car park
39.The mean of five numbers is 12. Four of the numbers are 10, 14, 11, and 13. What is the fifth number?
Medium
- A12
- B10
- C14
- D16
40.The range of a data set is 57. The lowest value is 39. What is the highest value?
Easy
41.For the same coin data (Value: 1,2,5,10,20,50; Frequency: 3,1,3,2,4,2), find the mode.
Easy
42.Complete the sentence.
Easy
The median is the ____ value when data is in order.
43.The table shows the time (t minutes) for 120 people in a supermarket. | Time (T min) | Frequency | |---|---| | 10 < T ≤ 30 | 16 | | 30 < T ≤ 40 | 18 | | 40 < T ≤ 45 | 22 | | 45 < T ≤ 50 | 40 | | 50 < T ≤ 60 | 21 | | 60 < T ≤ 70 | 3 | Use midpoints to estimate the mean time. Give your answer to 1 decimal place.
Medium
44.Find the median of the following numbers: 7, 3, 9, 5, 1.
Easy
45.Find the range of the numbers: 4, 7, 2, 9, 5.
Easy
46.Complete the sentence about data types.
Easy
The number of cars in a car park is an example of ____ data, while the weight of a car is an example of ____ data.
47.A class of 30 students has a mean test score of 72. After one student's score is added, the mean becomes 73. What is the score of the new student?
Medium
48.The table shows the times (t minutes) taken by 120 students to complete a puzzle. | Time (t min) | Frequency | |---|---| | 20 < t ≤ 25 | 44 | | 25 < t ≤ 30 | 32 | | 30 < t ≤ 35 | 28 | | 35 < t ≤ 40 | 12 | | 40 < t ≤ 45 | 4 | Estimate the mean time. Give your answer to 3 significant figures.
Medium
49.Explain why the median is a better measure of average than the mean when the data contains extreme values.
Medium
50.The table shows the time taken for each of 120 students to complete a puzzle. | Time (t min) | Frequency | |---|---| | 20 < t ≤ 25 | 44 | | 25 < t ≤ 30 | 32 | | 30 < t ≤ 35 | 28 | | 35 < t ≤ 40 | 12 | | 40 < t ≤ 45 | 4 | Write down the interval containing the median time.
Medium
51.The range of a data set is 57. The lowest value is 39. What is the highest value?
Easy
- A96
- B86
- C116
- D106
52.For the same coin data (Value: 1,2,5,10,20,50; Frequency: 3,1,3,2,4,2), find the mode.
Easy
- A18 cents
- B22 cents
- C20 cents
- D24 cents
53.Find the range of the numbers: 4, 7, 2, 9, 5.
Easy
- A7
- B9
- C8
- D6
54.A class of 30 students has a mean test score of 72. After one student's score is added, the mean becomes 73. What is the score of the new student?
Medium
- A93
- B113
- C123
- D103

Revision notes

## Averages & Range - Quick Revision ### Key Definitions - **Mean** = sum of values ÷ number of values. Use formula: total = mean × number. - **Median** = middle value when data is ordered. For n values, median is the ((n+1)/2)th value. - **Mode** = most frequent value. - **Range** = highest value - lowest value. - **Interquartile Range (IQR)** = Upper Quartile - Lower Quartile. Measures spread of middle 50%. ### Discrete vs Continuous Data - **Discrete**: specific values (e.g., shoe sizes, number of pets). - **Continuous**: any value on a scale (e.g., height, weight, time). ### Averages from Frequency Tables - **Mode**: value with highest frequency. - **Median**: find cumulative frequencies; locate ((n+1)/2)th value. - **Mean**: add column 'x × f', sum it, divide by total frequency. - **Range**: largest data value - smallest data value (not frequencies). ### Averages from Grouped Data - **Estimated Mean**: use midpoints of intervals. Formula: Σ(midpoint × frequency) ÷ total frequency. - **Modal Class**: class interval with highest frequency. - **Median Class**: find cumulative frequencies; locate ((n+1)/2)th value's interval. ### Comparing Data Sets - Compare **averages** (median/mean) and **spreads** (range/IQR). - Write: "[Group A] has a higher/lower [average] than [Group B], meaning ..." - Write: "[Group A] has a smaller/larger [spread] than [Group B], meaning ..." ### Important Tips - Mean is affected by outliers; median is better for skewed data. - For even number of values, median is midpoint of two middle values. - Quartiles: LQ = median of lower half (exclude median if odd n); UQ = median of upper half. - IQR = UQ - LQ; not affected by outliers.

Mean: The sum of all values divided by the number of values. Formula: mean = total of values ÷ number of values.
Median: The middle value when data is arranged in order. For n values, it is the ((n+1)/2)th value.
Mode: The value that appears most often in a data set.
Range: The difference between the highest and lowest values. Range = highest - lowest.
Interquartile Range (IQR): The difference between the upper quartile and the lower quartile. IQR = UQ - LQ. Measures spread of middle 50% of data.
Lower Quartile (LQ): The median of the lower half of the data. 25% of data lies below this value.
Upper Quartile (UQ): The median of the upper half of the data. 75% of data lies below this value.
Discrete Data: Numerical data that can only take specific values (often integers). E.g., number of pets, shoe sizes.
Continuous Data: Numerical data that can take any value within a range. E.g., height, weight, time.
Modal Class: For grouped data, the class interval with the highest frequency.
Estimated Mean: An approximation of the mean for grouped data, calculated using midpoints: Σ(midpoint × frequency) ÷ total frequency.

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