Averages And Range

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Notes

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 $= 1.56\times 24 =$ 37.44; total after $= 1.57\times 25 =$ 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 $\times frequency)$ , sum xf, divide by total frequency.
**Range**: difference between largest and smallest data values (not frequencies).

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.'

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.

Mean, Median & Mode on a Number Line

Finding Median from Frequency Table

Comparing Data Sets: Snails vs Slugs

Discrete vs Continuous Data Examples

Practice questions

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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.
Easy
- 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.
Hard
- 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.
Hard
- 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

Summary

## 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).

Key terms

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).

Slides

Mean, Median & Mode

The mode is the value that appears most often.
The median is the middle value when data is ordered.
The mean is the sum of values divided by the number of values.

For the data set 2, 3, 3, 5, 7, what is the mode?

Finding the Median

Order the data from smallest to largest.
If there is an odd number of values, the median is the middle one.
If there is an even number, the median is the midpoint of the two middle values.

For the data set 4, 1, 7, 2, the median $is ____$ .

Range

The range is the difference between the highest and lowest values.
Range = highest value – lowest value.
It measures how spread out the data is.

What is the range of 5, 9, 2, 7?

Choosing the Right Average

Use the mode for non-numerical data or when the most common value is needed.
Use the median when there are extreme values (outliers).
Use the mean when data is symmetrical and has no outliers.

Which of these are true?

The mean is affected by extreme values.
The median is always one of the data values.
The mode can be used for categorical data.

Calculations with the Mean

Mean = total of values ÷ number of values.
Total of values = mean × number of values.
Use this to find missing values or new totals when data changes.

The mean of 5 numbers is 12. The total of the numbers $is ____$ .

Averages from Frequency Tables

Mode: the data value with the highest frequency.
Median: the (n+1)/2th value, where n is total frequency.
Mean: sum of (data value $\times frequency)$ divided by total frequency.

From the table: x:1 f:3, x:2 f:5, x:3 f:2. What is the mode?

Comparing Data Sets

Compare averages (mode, median, mean) to see which set is higher/lower.
Compare ranges to see which set is more spread out.
Always link numbers to the context in your conclusion.

Which statements about comparing data are correct?

A higher median means better performance on average.
A larger range means data is more consistent.
The mean should be used if there are outliers.

Discrete & Continuous Data

Discrete data can only take particular values (e.g., number of people).
Continuous data can take any value on a scale (e.g., height, weight).
If you need a measuring device, it is likely continuous.

Which of these is discrete data?

Temperature
Number of cars
Height
Time

Flashcards

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 $\times frequency)$ 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 $= 16\times 7 = 112$ . 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 $= 1.56\times 24 = 37.44$ . Total after $= 1.57\times 25 = 39.25$ . 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 $= 0\times 14+1\times 0+2\times 2+3\times 5+4\times 11+5\times 8+6\times 10 = 0+0+4+15+44+40+60 = 163$ . Total customers $= 14+0+2+5+11+8+10 = 50$ . Mean $= \frac{163}{50} = 3.26$ .

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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