11.1 Analysing Data
Understanding Averages and Spread
Averages tell us about the typical value in a dataset. The range tells us how spread out the data is.
Averages tell us about the typical value in a dataset. The range tells us how spread out the data is.
⚡ The Four Key Statistics:
Mean
$$\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}$$
The arithmetic average
Median
Middle value when ordered
Not affected by outliers
Mode
Most frequent value
Can have more than one
Range
Highest - Lowest
Measures spread
Example 1: Finding All Four Statistics
Find the mean, median, mode, and range of: 3, 7, 5, 9, 5, 4, 5, 8
Mean:
Sum = 3 + 7 + 5 + 9 + 5 + 4 + 5 + 8 = 46
Number of values = 8
$$\text{Mean} = \frac{46}{8} = 5.75$$
Median:
First, order the data: 3, 4, 5, 5, 5, 7, 8, 9
Even number of values (8), so median is average of middle two
Middle values: 5 and 5
$$\text{Median} = \frac{5 + 5}{2} = 5$$
Mode:
5 appears most often (3 times)
Mode = 5
Range:
Highest value = 9, Lowest value = 3
Range = 9 - 3 = 6
Answers: Mean = 5.75, Median = 5, Mode = 5, Range = 6
Mean:
Sum = 3 + 7 + 5 + 9 + 5 + 4 + 5 + 8 = 46
Number of values = 8
$$\text{Mean} = \frac{46}{8} = 5.75$$
Median:
First, order the data: 3, 4, 5, 5, 5, 7, 8, 9
Even number of values (8), so median is average of middle two
Middle values: 5 and 5
$$\text{Median} = \frac{5 + 5}{2} = 5$$
Mode:
5 appears most often (3 times)
Mode = 5
Range:
Highest value = 9, Lowest value = 3
Range = 9 - 3 = 6
Answers: Mean = 5.75, Median = 5, Mode = 5, Range = 6
Example 2: Odd Number of Values
Find the median of: 12, 15, 9, 18, 11
Step 1: Order the data
9, 11, 12, 15, 18
Step 2: Find the middle value
5 values, so the middle is the 3rd value
Median = 12
Step 1: Order the data
9, 11, 12, 15, 18
Step 2: Find the middle value
5 values, so the middle is the 3rd value
Median = 12
Example 3: Finding a Missing Value
The mean of five numbers is 8. Four of the numbers are 6, 7, 9, and 10. Find the fifth number.
Solution:
Mean = 8 and there are 5 numbers
So: Sum ÷ 5 = 8
Therefore: Sum = 8 × 5 = 40
Sum of known numbers: 6 + 7 + 9 + 10 = 32
Fifth number = 40 - 32 = 8
Answer: The fifth number is 8
Solution:
Mean = 8 and there are 5 numbers
So: Sum ÷ 5 = 8
Therefore: Sum = 8 × 5 = 40
Sum of known numbers: 6 + 7 + 9 + 10 = 32
Fifth number = 40 - 32 = 8
Answer: The fifth number is 8
Example 4: Comparing Datasets
Two classes took a test:
Class A: Mean = 65, Range = 30
Class B: Mean = 65, Range = 10
Analysis:
• Both classes have the same average (mean = 65)
• Class B is more consistent (smaller range)
• Class A has more variation in scores
Conclusion: Class B performed more consistently, even though average scores were the same
Class A: Mean = 65, Range = 30
Class B: Mean = 65, Range = 10
Analysis:
• Both classes have the same average (mean = 65)
• Class B is more consistent (smaller range)
• Class A has more variation in scores
Conclusion: Class B performed more consistently, even though average scores were the same
⚡ When to Use Which Average:
| Average | Best For | Advantages | Disadvantages |
|---|---|---|---|
| Mean | Most situations | Uses all data; algebraically useful | Affected by outliers |
| Median | When there are outliers | Not affected by extreme values | Doesn't use all data |
| Mode | Categorical data | Easy to find; actual data value | May not exist or be unique |
💡 Statistics Tips:
• Mean: Add all values, divide by how many
• Median: ALWAYS order first, then find middle
• Mode: Can have more than one, or none at all
• Range: Always positive (or zero)
• Median: ALWAYS order first, then find middle
• Mode: Can have more than one, or none at all
• Range: Always positive (or zero)
🧮 Statistics Calculator (Do not use for question below... or do)
🎯 Some Practice (Statistics calculators very very forbidden)
Working with Frequency Tables
When data is organized in a frequency table, we can still calculate averages, but we need to account for how many times each value appears.
When data is organized in a frequency table, we can still calculate averages, but we need to account for how many times each value appears.
⚡ Frequency Table Formulas:
Mean from Frequency
$$\text{Mean} = \frac{\sum (x \times f)}{\sum f}$$
Sum of (value × frequency) ÷ Total frequency
Finding Position
Use cumulative frequency
Add frequencies to find position
Example 1: Mean from Frequency Table
The number of goals scored in 20 football matches:
Calculate mean:
$$\text{Mean} = \frac{\sum (x \times f)}{\sum f} = \frac{35}{20} = 1.75 \text{ goals}$$
Answer: Mean = 1.75 goals per match
| Goals (x) | Frequency (f) | x × f |
|---|---|---|
| 0 | 3 | 0 |
| 1 | 5 | 5 |
| 2 | 7 | 14 |
| 3 | 4 | 12 |
| 4 | 1 | 4 |
| Total | 20 | 35 |
$$\text{Mean} = \frac{\sum (x \times f)}{\sum f} = \frac{35}{20} = 1.75 \text{ goals}$$
Answer: Mean = 1.75 goals per match
Example 2: Median from Frequency Table
Using the same table, find the median:
Find median position:
Total = 20 values (even)
Median position = average of 10th and 11th values
Looking at cumulative frequency:
• 10th value is in the "2 goals" group (cumulative = 15)
• 11th value is also in the "2 goals" group
Median = 2 goals
| Goals | Frequency | Cumulative Frequency |
|---|---|---|
| 0 | 3 | 3 |
| 1 | 5 | 8 |
| 2 | 7 | 15 |
| 3 | 4 | 19 |
| 4 | 1 | 20 |
Total = 20 values (even)
Median position = average of 10th and 11th values
Looking at cumulative frequency:
• 10th value is in the "2 goals" group (cumulative = 15)
• 11th value is also in the "2 goals" group
Median = 2 goals
Example 3: Mode from Frequency Table
Using the same table:
Mode: The value with highest frequency
Looking at frequencies: 3, 5, 7, 4, 1
Highest frequency = 7 (for 2 goals)
Mode = 2 goals
Mode: The value with highest frequency
Looking at frequencies: 3, 5, 7, 4, 1
Highest frequency = 7 (for 2 goals)
Mode = 2 goals
Example 4: Range from Frequency Table
Using the same table:
Range: Highest value - Lowest value
Highest = 4 goals
Lowest = 0 goals
Range = 4 - 0 = 4 goals
Range: Highest value - Lowest value
Highest = 4 goals
Lowest = 0 goals
Range = 4 - 0 = 4 goals
💡 Frequency Table Tips:
• For mean: Multiply each value by its frequency first
• For median: Use cumulative frequency to find position
• For mode: Look for highest frequency
• Always check: Sum of frequencies = total number of values
• Show your working: Add an x×f column for calculations
• For median: Use cumulative frequency to find position
• For mode: Look for highest frequency
• Always check: Sum of frequencies = total number of values
• Show your working: Add an x×f column for calculations
🎯 Frequency Table Practice
Working with Grouped Data
When data is grouped into class intervals (like 0-10, 10-20), we can only estimate the mean using midpoints.
When data is grouped into class intervals (like 0-10, 10-20), we can only estimate the mean using midpoints.
⚡ Grouped Data Key Concepts:
Midpoint
$$\text{Midpoint} = \frac{\text{min} + \text{max}}{2}$$
Representative value for the group
Estimated Mean
$$\text{Mean} \approx \frac{\sum (m \times f)}{\sum f}$$
Use midpoints like values
Modal Class
Group with highest frequency
Can't find exact mode
Example 1: Estimated Mean from Grouped Data
The heights (in cm) of 50 students:
Calculate midpoints:
140-150: $(140 + 150) \div 2 = 145$
150-160: $(150 + 160) \div 2 = 155$
And so on...
Estimated mean:
$$\text{Mean} \approx \frac{8030}{50} = 160.6 \text{ cm}$$
Answer: Estimated mean ≈ 160.6 cm
| Height (cm) | Frequency (f) | Midpoint (m) | m × f |
|---|---|---|---|
| 140 ≤ h < 150 | 8 | 145 | 1160 |
| 150 ≤ h < 160 | 15 | 155 | 2325 |
| 160 ≤ h < 170 | 18 | 165 | 2970 |
| 170 ≤ h < 180 | 9 | 175 | 1575 |
| Total | 50 | 8030 |
140-150: $(140 + 150) \div 2 = 145$
150-160: $(150 + 160) \div 2 = 155$
And so on...
Estimated mean:
$$\text{Mean} \approx \frac{8030}{50} = 160.6 \text{ cm}$$
Answer: Estimated mean ≈ 160.6 cm
Example 2: Modal Class
Using the same height data:
Modal class: The group with the highest frequency
Frequencies: 8, 15, 18, 9
Highest frequency = 18
Modal class: 160 ≤ h < 170
Note: We can't find an exact mode from grouped data
Modal class: The group with the highest frequency
Frequencies: 8, 15, 18, 9
Highest frequency = 18
Modal class: 160 ≤ h < 170
Note: We can't find an exact mode from grouped data
Example 3: Median Class
Using the same height data, find the median class:
Find median position:
Total = 50 values
Median position = $\frac{50 + 1}{2} = 25.5$th value
Looking at cumulative frequency:
The 25.5th value is in the group where cumulative frequency reaches 25.5
This is 150 ≤ h < 160 (cumulative = 23 to 41)
Median class: 150 ≤ h < 160
| Height (cm) | Frequency | Cumulative Frequency |
|---|---|---|
| 140 ≤ h < 150 | 8 | 8 |
| 150 ≤ h < 160 | 15 | 23 |
| 160 ≤ h < 170 | 18 | 41 |
| 170 ≤ h < 180 | 9 | 50 |
Total = 50 values
Median position = $\frac{50 + 1}{2} = 25.5$th value
Looking at cumulative frequency:
The 25.5th value is in the group where cumulative frequency reaches 25.5
This is 150 ≤ h < 160 (cumulative = 23 to 41)
Median class: 150 ≤ h < 160
Example 4: Comparing Grouped Data
Two shops recorded customer wait times:
Shop A: Estimated mean = 8 minutes, modal class = 5-10 minutes
Shop B: Estimated mean = 8 minutes, modal class = 0-5 minutes
Analysis:
• Both have same average wait time
• Shop B's modal class is lower (most customers wait less)
• Shop B likely provides better service despite same average
Conclusion: Modal class gives additional insight beyond the mean
Shop A: Estimated mean = 8 minutes, modal class = 5-10 minutes
Shop B: Estimated mean = 8 minutes, modal class = 0-5 minutes
Analysis:
• Both have same average wait time
• Shop B's modal class is lower (most customers wait less)
• Shop B likely provides better service despite same average
Conclusion: Modal class gives additional insight beyond the mean
💡 Grouped Data Tips:
• Midpoint formula: (Lower + Upper) ÷ 2
• Median class: The group containing the median position
• Check intervals: Make sure they don't overlap
• Median class: The group containing the median position
• Check intervals: Make sure they don't overlap
🎯 Grouped Data Practice
Real Life Uses of Data Analysis:
• Business: Analyzing sales data, customer satisfaction
• Education: Comparing test scores, tracking progress
• Healthcare: Patient wait times, treatment effectiveness
• Sports: Player statistics, team performance
• Science: Experimental results, measurement data
• Government: Census data, economic indicators
• Business: Analyzing sales data, customer satisfaction
• Education: Comparing test scores, tracking progress
• Healthcare: Patient wait times, treatment effectiveness
• Sports: Player statistics, team performance
• Science: Experimental results, measurement data
• Government: Census data, economic indicators