Statistics organises scattered data into tables and graphs so you can see the whole picture at a glance.
Split the data into intervals and count how many values fall in each, and you have a frequency table; draw it as bars for a histogram, and join the midpoints of the bar tops for a frequency polygon.
Relative frequency, the share of the total, lets you compare two classes of different sizes fairly, and class marks let you estimate the mean even without the original data.
On this page you use the scores of 30 students and change the class width to see how the graphs and two versions of the mean change.
Organising 30 scores
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Histogram and frequency polygon of 30 scores: try changing the class width
10 pts
📊 Working with 30 scores
①We have the maths scores of 30 students
②Split the scores into equal intervals, such as 40 up to 50, 50 up to 60, and so on
③Counting how many students are in each interval gives a frequency table, and drawing it as bars gives a histogram
④Add an empty class at each end and join the midpoints of the bar tops: that is the frequency polygon
⑤Changing the class width changes the shape of the graph, even for the same data. Too narrow and it is jagged; too wide and the features blur
Frequency table vocabulary
Class mark
(sum of endpoints)2
The value that represents a class. For the class from 60 up to 70, it is (60 + 70) ÷ 2 = 65
🔍 Key vocabulary
①A data value is each individual value collected; here, one student's score
②A class is an interval of equal width, such as 60 up to 70
③The frequency is the number of values in a class
④The class width is the width of the interval: 70 − 60 = 10
⑤The class mark is the midpoint of a class; here, 65
Relative frequency
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Relative frequency of each class shown as a pie chart
Relative frequency
frequencytotal
The share of the total that a class makes up. The relative frequencies always add up to 1
💡 Why relative frequency?
①If our class has 30 students and the next has 35, raw frequencies make it hard to say which has more high scorers
②Dividing each frequency by the total gives the relative frequency, which compares fairly whatever the class sizes
③In this data, 12 students scored 80 or more, so the relative frequency is 12/30 = 0.4
④In a pie chart each slice is proportional to its relative frequency, and all the slices together make one whole circle: 1
The mean from a frequency table
Mean from a frequency table
Σ(class mark × freq)total
Without the original data, treat every value in a class as its class mark and estimate the mean
📝 Two versions of the mean
①Adding the 30 scores and dividing by 30 gives a mean of about 75.4
②With a class width of 10, the class marks give (45 × 1 + 55 × 3 + 65 × 6 + 75 × 8 + 85 × 6 + 95 × 6) ÷ 30 = 76
③The two differ slightly because every value in a class was replaced by its class mark
④Change the class width in the picture and the mean from class marks changes too
Check with examples
Example 1
Find the mean of the data 4, 6, 8, 10, 7.
1
Mean = (sum of the data) ÷ (number of values).
(4 + 6 + 8 + 10 + 7) ÷ 5
2
Calculate.
= 35 ÷ 5 = 7
▸ 7
The amounts above the mean, (8 − 7) + (10 − 7) = 4, match the amounts below it, (7 − 4) + (7 − 6) = 4. The mean is the balance point of the data.
Example 2
A set of data has total frequency 20, and one class has frequency 5. Find that class's relative frequency.
1
Relative frequency = (frequency of the class) ÷ (total frequency).
relative frequency = 5 ÷ 20
2
Calculate.
= 0.25
▸ 0.25
A relative frequency of 0.25 means 25% of the total; in a pie chart it fills a quarter of the circle.
Exam checklist
Key formulas
relative frequency = frequency ÷ total, mean ≈ Σ(class mark × frequency) ÷ total
Relative frequencies add up to 1, and the mean from class marks is an estimate
Grade 7 school test style
The mean of the four numbers 3, 7, x, 10 is 7. What is x?
①6
②7
③8
④9
⑤10
▸ ③ 8
1
Write an equation from the definition of the mean.
(3 + 7 + x + 10) ÷ 4 = 7
2
Multiply both sides by 4 and solve for x.
20 + x = 28 ⇒ x = 8
🎯 Exam checklist
①A class mark is the sum of the class's two end values divided by 2
②All the relative frequencies add up to 1
③The total area of the histogram bars is (class width) × (total frequency), the same as the area between the frequency polygon and the horizontal axis
④Changing the class width changes the shape of the graph, even for the same data
⑤A mean found from a frequency table may differ slightly from the mean of the original data