Averages and range

1. Overview

Averages and range are fundamental tools in statistics for understanding and comparing data. Averages (mean, median, and mode) pinpoint a typical or central value within a dataset, while the range quantifies the spread or variability of the data. Mastering these concepts allows you to effectively summarize data, identify trends, and make informed comparisons. This revision guide covers how to calculate and interpret the mean, median, mode, and range for individual data sets, equipping you with the skills needed for the IGCSE Cambridge Mathematics (0580) exam.

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Key Definitions

• Mean: The sum of all values divided by the total number of values.
• Median: The middle value when the data is arranged in numerical order.
• Mode: The value that appears most frequently in a dataset.
• Range: The difference between the highest and lowest values in a dataset (a measure of spread, not an average).
• Outlier: A value that is significantly higher or lower than the rest of the data, which can distort the mean.

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Core Content

The Mode

The mode is the only average that can be used for non-numerical data (e.g., favorite colors).

• Note: A dataset can have one mode, be bimodal (two modes), or have no mode if all values appear once.

The Median

To find the median, you must order the data from smallest to largest first.

• Position formula: $\frac{n + 1}{2}$, where $n$ is the number of values.
• If $n$ is odd, the median is a specific number in the list.
• If $n$ is even, the median is the mean of the two middle numbers.

Worked example 1 — Finding the Median

Question: Find the median of the following set of numbers: $15, 6, 12, 9, 18, 10, 15$.

1. Order the data: $6, 9, 10, 12, 15, 15, 18$ Reason: Arranging the data in ascending order is necessary to identify the middle value.
2. Count the values ($n$): $n = 7$ Reason: We need to know the total number of values to apply the median position formula.
3. Find the position: $\frac{7 + 1}{2} = 4^{\text{th}}$ position. Reason: This formula tells us which value in the ordered list represents the median.
4. Identify the median: The $4^{\text{th}}$ value in the ordered list is $12$.

Median = 12

The Mean

The mean is the most commonly used average but is sensitive to outliers.

Worked example 2 — Finding the Mean

Question: Calculate the mean height of five students whose heights are $155\text{ cm}, 160\text{ cm}, 162\text{ cm}, 158\text{ cm},$ and $165\text{ cm}$.

1. Sum the values: $155 + 160 + 162 + 158 + 165 = 800$ Reason: The mean is the sum of all values divided by the number of values.
2. Divide by $n$ ($5$ values): $\frac{800}{5} = 160$ Reason: Dividing the sum by the number of values gives us the mean.
3. Final Answer: $\boxed{160\text{ cm}}$

Worked example 3 — Reverse Mean

Question: The mean weight of 6 boxes is 8 kg. Five of the boxes have weights of 5 kg, 7 kg, 9 kg, 10 kg, and 11 kg. What is the weight of the sixth box?

1. Calculate the total weight of all 6 boxes: $6 \times 8 = 48$ kg Reason: Since mean = (total weight) / (number of boxes), then total weight = mean × (number of boxes).
2. Calculate the total weight of the 5 known boxes: $5 + 7 + 9 + 10 + 11 = 42$ kg Reason: We need to find the combined weight of the boxes we already know about.
3. Subtract the weight of the 5 boxes from the total weight of all 6 boxes: $48 - 42 = 6$ kg Reason: The difference between the total weight and the weight of the known boxes is the weight of the missing box.
4. Final Answer: $\boxed{6 \text{ kg}}$

The Range

The range indicates how consistent the data is. A small range means the data is consistent; a large range means the data is spread out.

• Calculation: $\text{Highest Value} - \text{Lowest Value}$

Worked example 4 — Finding the Range

Question: Find the range of the following temperatures recorded in a city over a week: $22°C, 25°C, 21°C, 28°C, 23°C, 20°C, 24°C$.

1. Identify the highest value: $28°C$ Reason: The range is calculated by subtracting the lowest value from the highest value.
2. Identify the lowest value: $20°C$ Reason: We need to find the smallest temperature recorded.
3. Calculate the range: $28 - 20 = 8$ Reason: Subtracting the lowest value from the highest value gives us the range.
4. Final Answer: $\boxed{8°C}$ Interpretation: The temperatures varied by 8°C over the week.

Choosing the Best Average

• Use Mode when the data is non-numerical (e.g., most popular car brand).
• Use Median when there are extreme outliers (e.g., house prices or salaries).
• Use Mean when you want to include every piece of data and there are no extreme outliers.

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Extended Content (Extended Only)

Extended students are expected to calculate the estimated mean from grouped frequency tables. This involves using the midpoint of each class interval as a representative value for that interval. The estimated mean provides an approximation of the true mean when the raw data is not available, only grouped data.

Worked Example 5 — Estimated Mean from Grouped Data

Question: The table below shows the distribution of the heights of 50 students in a school. Calculate an estimate for the mean height.

Height (cm)	Frequency
$150 < h \leq 160$	8
$160 < h \leq 170$	15
$170 < h \leq 180$	17
$180 < h \leq 190$	10

1. Find the Midpoints ($x$):

   • For $150 < h \leq 160$, midpoint $= \frac{150 + 160}{2} = 155$
   • For $160 < h \leq 170$, midpoint $= \frac{160 + 170}{2} = 165$
   • For $170 < h \leq 180$, midpoint $= \frac{170 + 180}{2} = 175$
   • For $180 < h \leq 190$, midpoint $= \frac{180 + 190}{2} = 185$ Reason: The midpoint represents the average value within each height interval.

2. Multiply $f \times x$:

   • $8 \times 155 = 1240$
   • $15 \times 165 = 2475$
   • $17 \times 175 = 2975$
   • $10 \times 185 = 1850$ Reason: Multiplying the frequency by the midpoint gives the total "height" for each group.

3. Sum of $fx$: $1240 + 2475 + 2975 + 1850 = 8540$ Reason: Summing the products gives the total estimated "height" of all students.

4. Sum of $f$: $8 + 15 + 17 + 10 = 50$ Reason: This is the total number of students.

5. Estimated Mean: $\frac{8540}{50} = 170.8\text{ cm}$ Reason: Dividing the total estimated "height" by the total number of students gives the estimated mean height.

Final Answer: $\boxed{170.8 \text{ cm}}$

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Key Equations

• Mean: $\bar{x} = \frac{\sum x}{n}$

  • $\sum x$ = sum of all values
  • $n$ = number of values
  • Note: Not on formula sheet - must memorise.

• Range: $x_{max} - x_{min}$

  • Note: Not on formula sheet - must memorise.

• Median Position: $\frac{n + 1}{2}$

  • Note: Not on formula sheet - must memorise.

• Estimated Mean: $\frac{\sum (f \times x)}{\sum f}$ (where $x$ is the class midpoint)

  • Note: Not on formula sheet - must memorise.

Note: These formulas are NOT provided on the IGCSE formula sheet. You must memorise them.

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Common Mistakes to Avoid

• ❌ Wrong: Calculating the median of the numbers $4, 7, 2, 9, 1$ as $7$ (the middle number in the unordered list). ✓ Right: Always rewrite the list from smallest to largest first: $1, 2, 4, 7, 9$. The median is $4$.
• ❌ Wrong: Interpreting a stem-and-leaf diagram with key $1|5 = 1.5$ as meaning the value is fifteen, rather than one point five. ✓ Right: Always check the key; $1|5$ might mean $1.5$ or $15$. Read carefully!
• ❌ Wrong: Stating "the range of temperatures was from $15°C$ to $25°C$." ✓ Right: The range is a single value ($25°C - 15°C = 10°C$). The range was $10°C$.
• ❌ Wrong: When calculating the mean of 7 numbers, accidentally adding only 6 of them, leading to an incorrect sum and therefore an incorrect mean. ✓ Right: Tick off each number as you add it to ensure $n$ is correct and you've included every data point.
• ❌ Wrong: When finding the median of an even number of values, forgetting to average the two middle values. For example, stating the median of $2, 4, 6, 8$ is $4$ or $6$ instead of $5$. ✓ Right: With an even number of values, identify the two middle numbers and calculate their mean. In this case, $(4+6)/2 = 5$.

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Exam Tips

• Comparison Questions: If asked to "Compare two sets of data," you must comment on one average (usually mean or median) AND the spread (range).
  • Example: "Class A had a higher mean score (higher performance), but Class B had a smaller range (more consistent performance)."
• Calculator Tip: When calculating the mean on a calculator, press = after summing the numbers and before dividing. If you type $2 + 4 + 6 / 3$, the calculator will do $2 + 4 + (6/3) = 8$ instead of $(2+4+6)/3 = 4$. Use brackets to ensure correct order of operations: $(2+4+6)/3$.
• Reverse Mean Questions: If you are given the mean and asked to find a missing value, multiply the mean by $n$ to find the total sum first.
• Context: If a question asks why the median is better than the mean, the answer is almost always "because the data contains outliers/extreme values that would distort the mean."