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Averages and Range from Tables

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Foundational skills
Read a frequency
\[ \text{How many scored } 3? \]
Read how often a value appears from the table.
Total frequency
\[ \Sigma f = 4+7+5+2 \]
Add the frequencies to find how many pieces of data there are.
Total of the data (Σfx)
\[ \Sigma fx = 0\times4+1\times7+\ldots \]
Multiply each value by its frequency, then add.
Missing frequency, given the total
\[ 4+x+5+2=20 \]
Use the total frequency to find a missing frequency.
Midpoint of a class (inequalities)
\[ 20 \lt x \le 30 \to 25 \]
Find the midpoint of a class like 20 < x ≤ 30.
Midpoint of a discrete class
\[ 10\text{–}14 \to 12 \]
Find the midpoint of a class like 10–14.
Mode and range
Mode
\[ \text{Mode} = ? \]
Find the value with the highest frequency.
Range
\[ 6-1=5 \]
Largest value minus smallest value.
Range when an end value has frequency 0
\[ \text{ignore } f=0 \]
Ignore values with a frequency of 0 when finding the range.
Median
Position of the median
\[ \frac{25+1}{2}=13\text{th} \]
Use (n + 1) ÷ 2 to find where the median is.
Median, odd total
\[ \text{13th value} \]
Find the median when there is an odd number of values.
Median, even total, same group
\[ \text{10th and 11th} \]
The two middle values are the same.
Median, even total, across two groups
\[ \frac{3+4}{2}=3.5 \]
The two middle values are in different columns.
Mean
Mean, exact answer
\[ \frac{\Sigma fx}{\Sigma f}=\frac{60}{20} \]
Divide Σfx by Σf. The answer is exact.
Mean to 1 d.p.
\[ \frac{47}{15}=3.1 \]
Divide Σfx by Σf and round to 1 decimal place.
Mean when 0 is a value
\[ 0 \text{ still counts} \]
Values of 0 still count in Σf.
Grouped: classes
Modal class
\[ 20 \lt x \le 30 \]
Find the class with the highest frequency.
Class containing the median
\[ \frac{40+1}{2}=20.5\text{th} \]
Use (n + 1) ÷ 2 and a running total to find the median class.
Greatest possible range
\[ 50-10=40 \]
Top of the last class minus the bottom of the first class.
Grouped: estimating
Estimate the total (Σfm)
\[ \Sigma fm \]
Multiply each midpoint by its frequency, then add.
Estimate the mean, exact answer
\[ \frac{\Sigma fm}{\Sigma f} \]
Divide Σfm by Σf. The answer is exact.
Estimate the mean to 1 d.p.
\[ \bar{x}\approx 23.7 \]
Divide Σfm by Σf and round to 1 decimal place.
Estimate the mean, unequal widths
\[ m = 5,\ 15,\ 35 \]
Work out each midpoint separately when the classes are different widths.
Extensions
Missing frequency, given the mean
\[ \frac{30+2x}{10+x}=2.5 \]
Form and solve an equation using the mean.
Missing frequency, given the estimated mean
\[ \text{est. mean } 24 \Rightarrow x? \]
Form and solve an equation using midpoints and the estimated mean.
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