Averages from tables 2 Revision Notes for AQA GCSE Maths

Averages from tables 2

What are frequency tables?

A frequency table shows how often different values appear in a dataset. The frequency tells you how many times each value occurs, not the value itself. This is crucial when calculating averages - you must be careful not to confuse the frequency with the actual data values.

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The frequency represents the count of how many times each value appears, while the value itself is the actual data point. This distinction is essential for accurate calculations.

Key measures of average

There are several types of average you can calculate from frequency tables:

Mode - the most frequently occurring value
Median - the middle value when all data is arranged in order
Mean - the total of all values divided by the number of values
Range - the difference between the highest and lowest values (measure of spread)

Finding the mode

The mode is the value that appears most often in the dataset. To find it from a frequency table, look for the value with the highest frequency.

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Worked Example: Finding the Mode

Table

Looking at the pet ownership data:

0 pets: frequency = 13
1 pet: frequency = 18 (highest)
2 pets: frequency = 7
3 pets: frequency = 2

Since 1 pet has the highest frequency (18), the mode is 1 pet.

Calculating the range

The range shows how spread out the data is. Calculate it by subtracting the smallest value from the largest value.

$\text{Range} = \text{Highest value} - \text{Lowest value}$

Using the pet ownership data:

$\text{Range} = 3 - 0 = 3 \text{ pets}$

Working out the median

The median is the middle value when all data points are arranged in order. For frequency tables, follow these steps:

Find the total number of data points by adding all frequencies
The median position is at $(n + 1) ÷ 2$ , where $n$ is the total frequency
Count through the frequencies to find which value is at this position

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Worked Example: Finding the Median

For the pet ownership data:

Step 1: Calculate total frequency Total frequency = $13 + 18 + 7 + 2 = 40$ people

Step 2: Find median position Median position = $(40 + 1) ÷ 2 = 20.5$ This means the median lies between the 20th and 21st values.

Step 3: Locate the position

Values 1-13: 0 pets
Values 14-31: 1 pet (includes both 20th and 21st positions)

Since both the 20th and 21st values are 1 pet, the median is 1 pet.

Calculating the mean from frequency tables

The mean requires special care with frequency tables. You cannot just add the values and divide - you must account for how often each value occurs.

Step-by-step method:

Create an extra column headed "frequency × value" $(f × x)$
Multiply each value by its frequency
Add up all the $f × x$ values to get the total
Divide this total by the sum of all frequencies

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Formula for Mean from Frequency Tables:

$\text{Mean} = \frac{\text{Total of } f × x \text{ column}}{\text{Total frequency}}$

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Worked Example: Calculating the Mean

Table

For the goals scored data:

Step 1: Create the $f × x$ column

$0 × 7 = 0$
$1 × 15 = 15$
$2 × 14 = 28$
$3 × 7 = 21$
$4 × 4 = 16$
$5 × 3 = 15$

Step 2: Calculate totals Total of $f × x$ column: $0 + 15 + 28 + 21 + 16 + 15 = 95$ Total frequency: $7 + 15 + 14 + 7 + 4 + 3 = 50$

Step 3: Apply the formula Mean = $\frac{95}{50} = 1.9$ goals

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Critical Points for Exam Success:

Always create the $f × x$ column when calculating the mean
Do not round your final answer unless specifically asked
Check your arithmetic carefully - it's easy to make errors with multiple calculations
Remember that frequency tells you how many, not what the value is
Show your working clearly for full marks

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Common Mistakes to Avoid:

Using frequency values as data values in calculations
Forgetting to create the $f × x$ column for mean calculations
Rounding too early in calculations
Mixing up mode (most frequent) with median (middle value)
Confusing the frequency with the actual data values

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Key Points to Remember:

Mode = value with the highest frequency
Range = highest value - lowest value
Median = middle value when data is in order
Mean from frequency tables = $\frac{\text{total of } f × x \text{ column}}{\text{total frequency}}$
Always create the $f × x$ column when finding the mean from a frequency table

Averages from tables 2 (AQA GCSE Maths): Revision Notes