Averages And Range

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Notes de leçon

Mean, Median & Mode

Mode: value that appears most often; e.g., mode of 1,2,2,5,6 is 2. There can be more than one mode.
Median: middle value when data are 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 fraction or decimal.
Use mode for non‑numerical data; avoid mean if there are extreme values (outliers); avoid mode if more than one mode.

Calculations with the Mean

Mean = total of values ÷ number of values. Rearranged: total = mean × number of values.
To find a missing value, compare totals before and after adding/removing a value.
Example: 24 students mean height 1.56 m; new student raises mean to 1.57 m. Total before = 1.56×24 = 37.44; total after = 1.57×25 = 39.25; new student height = 39.25 – 37.44 = 1.81 m.

Range

Range = highest value – lowest value; measures spread.
Affected by extreme values (outliers); do not use if outliers exist.
Always show subtraction clearly in working.

Averages from Tables

Mode from frequency table: data value with highest frequency.
Median: find the (n+1)/2 th value using cumulative frequencies.
Mean: add column xf (data value × frequency), sum xf, divide by total frequency.
Range: difference between largest and smallest data values (not frequencies).

Number of pets per house

Comparing Data Sets

Compare averages (mode/median/mean) and spread (range).
Conclusion must include numerical comparison and real‑life meaning.
Avoid mean if data has extreme values.
Example: 'Snails median 7.1 cm < slugs median 9.7 cm, so on average slugs travel further.'

Distance travelled in 10 minutes

Discrete & Continuous Data

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

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1.Find the mode of these numbers: 3, 4, 1, 3, 2, 1, 3, 4, 2, 2, 1, 3.
Easy
- A1
- B2
- C3
- D4
2.Find the median of these numbers: 27, 14, 8, 93, 32, 55, 14, 38, 73, 47.
Medium
- A32
- B35
- C38
- D14
3.Calculate the range of these numbers: 27, 14, 8, 93, 32, 55, 14, 38, 73, 47.
Easy
- A85
- B86
- C93
- D79
4.What is the mode of these numbers? 18, 13, 15, 8, 9, 17, 12, 8, 6, 14.
Easy
- A8
- B13
- C14
- D18
5.The table shows the number of bananas bought by 50 customers. Find the median. | Number of bananas | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---|---| | Frequency | 14 | 0 | 2 | 5 | 11 | 8 | 10 |
Medium
- A3
- B4
- C5
- D2
6.The mean of seven numbers is 16. Six of the numbers are 12, 20, 19, 10, 21 and 13. Find the seventh number.
Medium
- A17
- B18
- C19
- D16
7.Five numbers have a mean of 9.4. Four of the numbers are 3, 5, 10 and 12. Work out the range of the five numbers.
Medium
- A13
- B14
- C15
- D12
8.Which type of average (mean, median, or mode) can be used for non-numerical data such as methods of travel?
Easy
- AMean
- BMedian
- CMode
- DAll three

Plus de questions

9.State whether the data 'the weights of dogs' is discrete or continuous.
Easy
- ADiscrete
- BContinuous
- CBoth
- DNeither
10.State whether the data 'the number of computers owned by each household' is discrete or continuous.
Easy
- ADiscrete
- BContinuous
- CBoth
- DNeither
11.The mean of three numbers is 150. The numbers are 361, n, and (n-1). Find the value of n.
Medium
- A45
- B46
- C44
- D47
12.The table shows the number of examinations taken by 196 candidates. Find the median number of examinations. | Number of examinations | 5 | 6 | 7 | 8 | |---|---|---|---|---| | Number of candidates | 27 | 42 | 63 | 64 |
Medium
- A6
- B7
- C8
- D5
13.In its first 20 games, a basketball team's mean score was 76.25 points per game. In the next 12 games, the mean score was 80.25 points per game. Calculate the mean for all 32 games.
Medium
- A77.75
- B78.25
- C78.0
- D77.5
14.The stem-and-leaf diagram shows the scores of 25 students in a test. | Stem | Leaf | |---|---| | 5 | 2 6 8 | | 6 | 1 4 5 7 9 | | 7 | 0 1 2 3 4 5 6 8 9 | | 8 | 0 1 3 5 6 8 | | 9 | 0 2 | Key: 5 | 2 means a score of 52. Find the median score.
Medium
- A73
- B74
- C75
- D76
15.Find the range of the times: 12.5, 14.9, 9.0, 12.1, 14.4, 12.9, 10.0, 7.1, 11.0, 9.4 (in seconds).
Easy
- A7.8
- B7.9
- C7.4
- D8.1
16.The table shows the number of letters in each word for 25 words. Calculate the mean. | Number of letters | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---| | Number of words | 3 | 1 | 5 | 8 | 6 | 2 |
Medium
- A3.68
- B3.72
- C3.64
- D3.76
17.Find the median of these numbers: 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 17, 19, 20, 21, 23.
Easy
- A14
- B15
- C14.5
- D13.5
18.Find the mode of these numbers: 1, 2, 2, 5, 5, 6.
Easy
- A2
- B5
- C2 and 5
- D1
19.Calculate the mean of these numbers: 1, 2, 6.
Easy
- A3
- B2
- C4
- D3.5
20.What is the range of these numbers: 1, 2, 5, 8?
Easy
- A7
- B6
- C8
- D5
21.Find the median of these numbers: 1, 2, 3, 4.
Easy
- A2.5
- B2
- C3
22.The table shows the number of pets in 20 houses. Find the median. | Number of pets | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | Frequency | 2 | 7 | 6 | 4 | 1 |
Medium
- A2
- B1
- C3
- D1.5
23.The table shows the number of pets in 20 houses. Calculate the mean. | Number of pets | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | Frequency | 2 | 7 | 6 | 4 | 1 |
Medium
- A1.75
- B2
- C1.5
- D2.25
24.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.
Hard
- A1.81 m
- B1.80 m
- C1.82 m
- D1.79 m
25.Find the median of these numbers: 12, 10, 15, 14, 17, 11, 12, 13, 9, 21, 14, 20, 19, 16, 23.
Medium
- A14
- B15
- C16
- D13
26.Calculate the mean of these numbers: 10, 15, 12, 20, 13.
Easy
- A14
- B15
- C13
- D16
27.The table shows the number of pets in 20 houses. Find the mode. | Number of pets | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | Frequency | 2 | 7 | 6 | 4 | 1 |
Easy
- A1
- B2
- C7
- D0
28.The range is a measure of average.
Easy
True or false?
29.Which average is most appropriate for non-numerical data such as favourite colour?
Easy
- AMean
- BMedian
- CMode
- DRange
30.For the data set: 3, 7, 7, 10, 12, which of the following is true?
Medium
- AMean = 7.8, Median = 7, Mode = 7
- BMean = 7.8, Median = 7, Mode = 7.8
- CMean = 7, Median = 7.8, Mode = 7
- DMean = 7, Median = 7, Mode = 7.8
31.Match each average to its correct description.
Easy
- Mean
- Median
- Mode
- Range
- Middle value when ordered
- Most frequent value
- Sum divided by count
- Highest minus lowest
32.Arrange these steps to find the median from a frequency table in the correct order.
Medium
- Find the total frequency n.
- Calculate (n+1)/2 to locate the position.
- Read the data value at that position from the cumulative frequencies.
- Add a cumulative frequency column.
33.Which of the following are measures of central tendency? (Select all that apply.)
Easy
- AMean
- BRange
- CMedian
- DMode
- EVariance
34.The mean of five numbers is 12. Four of the numbers are 10, 14, 11, and 13. What is the fifth number?
Medium
- A10
- B12
- C14
- D16
35.Two classes took the same test. Class A had a median score of 65 and a range of 20. Class B had a median score of 72 and a range of 30. Which statement best compares the classes?
Medium
- AClass A performed better and was more consistent.
- BClass B performed better but was less consistent.
- CClass A performed worse and was less consistent.
- DClass B performed worse but was more consistent.
36.What is the mode of the following numbers? 2, 3, 5, 3, 4, 3, 6
Easy
37.What is the median of these numbers? 5, 2, 8, 1, 9
Easy
38.Calculate the mean of 4, 7, 9.
Easy
39.Find the range of these numbers: 12, 5, 20, 8, 15.
Easy
40.State whether the data 'the number of cars in a car park' is discrete or continuous.
Easy
41.State whether the data 'the time taken to run 100 metres' is discrete or continuous.
Easy
42.The mean of five numbers is 12. If four of the numbers are 10, 14, 11 and 13, find the fifth number.
Medium
43.The table shows the number of siblings of 20 students. Find the mean number of siblings. | Number of siblings | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | Frequency | 4 | 8 | 5 | 2 | 1 |
Medium
44.Which average (mean, median, or mode) is most appropriate for non-numerical data such as favourite colour?
Easy
45.Find the range of the following numbers: 22, 17, 12, 7, 2.
Easy
46.Complete the sentence.
Easy
The mode is the value that appears the ____ often.
47.Two classes took the same test. Class A had a median score of 65 and a range of 20. Class B had a median score of 72 and a range of 30. Compare the performance of the two classes.
Medium
48.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.
Hard
49.Calculate the mean of 4, 7, 9.
Easy
- A5.6667
- B8.6667
- C6.6667
- D7.6667
50.Find the range of these numbers: 12, 5, 20, 8, 15.
Easy
- A17
- B16
- C15
- D14
51.The table shows the number of siblings of 20 students. Find the mean number of siblings. | Number of siblings | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | Frequency | 4 | 8 | 5 | 2 | 1 |
Medium
- A1.4
- B3.4
- C0.4
- D2.4
52.Find the range of the following numbers: 22, 17, 12, 7, 2.
Easy
- A22
- B20
- C24
- D18

Notes de révision

## Averages & Range - Quick Revision ### Mean, Median & Mode - **Mode**: Most frequent value. Can have >1 mode. Non-numerical data → only mode works. - **Median**: Middle value when ordered. For even count, average the two middle numbers. - **Mean**: Sum of values ÷ number of values. Affected by extreme values (outliers). Avoid mean if outliers exist. ### Calculations with the Mean - Formula: mean = total ÷ number of values - Rearranged: total = mean × number of values - Use totals before and after to find missing values. ### Range - Range = highest value – lowest value - Measures spread. Affected by outliers. ### Averages from Tables - **Mode**: Data value with highest frequency. - **Median**: (n+1)/2 th value. Use cumulative frequency to locate. - **Mean**: mean = Σ(xf) / Σf (add column x×f, divide by total frequency). - **Range**: largest data value – smallest data value (not frequencies). ### Comparing Data Sets - Compare averages (median/mean) and spread (range). - Write: "[Group A] has a higher [average] than [Group B], meaning [context]." - Write: "[Group A] has a lower range, so scores are more consistent." ### Discrete & Continuous Data - **Discrete**: Specific values (e.g., shoe sizes, number of people). - **Continuous**: Any value on a scale (e.g., height, weight, time). ### Key Points - Outliers → use median, not mean. - Mean can be a decimal/fraction. - Range = highest – lowest (show subtraction).

Mean: Sum of all values divided by the number of values. Formula: mean = total ÷ n.
Median: The middle value when data is ordered. For even n, it is the average of the two middle values.
Mode: The value that appears most frequently. There can be more than one mode.
Range: The difference between the highest and lowest values. Range = highest – lowest.
Outlier: An extreme value that differs significantly from other values. Outliers affect the mean but not the median.
Frequency table: A table showing data values and how often each occurs (frequency). Used to find averages and range.
Discrete data: Data that can only take specific numerical values, often integers (e.g., number of pets).
Continuous data: Data that can take any numerical value on a scale (e.g., height, weight).

Cartes mémo

What is the mode?

The mode is the value that appears most often. There can be more than one mode.

How do you find the median for an odd number of values?

Order the numbers and choose the middle value. For example, the median of 2, 3, 4 is 3.

How do you find the median for an even number of values?

Order the numbers and find the midpoint of the two middle values. For example, the median of 1, 2, 3, 4 is (2+3)/2 = 2.5.

What is the mean?

The mean is the sum of the values divided by the number of values. For example, the mean of 1, 2, 6 is (1+2+6)/3 = 3.

What is the range?

The range is the difference between the highest and lowest values. For example, the range of 1, 2, 5, 8 is 8 - 1 = 7.

When should you avoid using the mean as an average?

Avoid the mean when there are extreme values (outliers) because it is badly affected by them.

When should you avoid using the mode as an average?

Avoid the mode when there is more than one mode, as it is not clear.

What type of average can be used for non-numerical data?

Only the mode can be used for non-numerical data.

How do you find the total of values from the mean?

Total of values = mean × number of values.

How do you find the mode from a frequency table?

The mode is the data value with the highest frequency.

How do you find the median from a frequency table?

Find the (n+1)/2th value, where n is the total frequency. Use cumulative frequencies to locate the data value.

How do you find the mean from a frequency table?

Mean = total of (data value × frequency) divided by total frequency.

How do you find the range from a frequency table?

Range = largest data value - smallest data value (not the difference of frequencies).

What is discrete data?

Discrete data can only take particular numerical values, often integers. Examples: number of people, shoe sizes.

What is continuous data?

Continuous data can take any numerical value on a scale. Examples: height, weight, length.

How do you compare two data sets?

Compare their averages (mode, median, or mean) and their spreads (range). Write what the numbers mean in real life.

What does a lower range indicate about a data set?

A lower range indicates that the data values are closer together, more consistent, or have less variation.

Find the median of: 280, 200, 175, 1180, 95, 182, 238, 256, 194, 250.

Ordered: 95, 175, 182, 194, 200, 238, 250, 256, 280, 1180. Median = (200+238)/2 = 219.

Find the range of times: 12.5, 14.9, 9.0, 12.1, 14.4, 12.9, 10.0, 7.1, 11.0, 9.4.

Highest = 14.9, lowest = 7.1, range = 14.9 - 7.1 = 7.8 s.

Find the median of: 27, 14, 8, 93, 32, 55, 14, 38, 73, 47.

Ordered: 8, 14, 14, 27, 32, 38, 47, 55, 73, 93. Median = (32+38)/2 = 35.

The mean of seven numbers is 16. Six numbers are 12, 20, 19, 10, 21, 13. Find the seventh number.

Total = 16×7 = 112. Sum of six = 12+20+19+10+21+13 = 95. Seventh = 112-95 = 17.

A class of 24 students has mean height 1.56 m. A new student joins, mean becomes 1.57 m. Find the new student's height.

Total before = 1.56×24 = 37.44. Total after = 1.57×25 = 39.25. Height = 39.25-37.44 = 1.81 m.

Find the mean from the frequency table: Number of bananas: 0(14), 1(0), 2(2), 3(5), 4(11), 5(8), 6(10).

Total bananas = 0×14+1×0+2×2+3×5+4×11+5×8+6×10 = 0+0+4+15+44+40+60 = 163. Total customers = 14+0+2+5+11+8+10 = 50. Mean = 163/50 = 3.26.

Compare snails (median 7.1 cm, range 3.1 cm) and slugs (median 9.7 cm, range 4.5 cm).

Slugs have a higher median (9.7 > 7.1), so on average slugs travel further. Snails have a lower range (3.1 < 4.5), so snail distances are less spread out.

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