Data Interpretation (Grouped Data)
Data Handling and Probability · Data Interpretation (Grouped Data)
Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 5.0 Data Handling and Probability | Sub-Strand 5.1 Data Interpretation, Grouped Data (6 Lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Construct a grouped frequency distribution table from raw data.
- Determine class boundaries, class width and class midpoints.
- Calculate the mean of grouped data and identify the modal and median classes.
- Draw and interpret histograms and frequency polygons.
Data Interpretation
A teacher marking forty scripts does not want a list of forty numbers. She wants to know how the class performed as a whole.
Grouping the marks into classes makes the pattern visible. This topic is about reading that pattern.
a) Grouped frequency tables
When there are many values, we sort them into classes and count how many fall in each. That count is the frequency.
| Marks | Frequency |
|---|---|
| 1-20 | 4 |
| 21-40 | 7 |
| 41-60 | 13 |
| 61-80 | 10 |
| 81-100 | 6 |
The frequencies add up to 40, which is the number of learners. Always check that total before going further.
Grouping costs you something. Once the marks are in classes, you no longer know any learner's exact score. You trade detail for a clearer overall picture.
b) Class midpoints
The exact values are gone. So we use the middle of each class to stand for every value in it.
c) The modal class
The modal class is simply the class with the highest frequency.
Here it is 41-60, with 13 learners. No calculation is needed, only reading.
Note that we say modal class, not mode. We cannot name a single most common mark, because the grouping hid the individual values.
d) The mean of grouped data
Multiply each midpoint by its frequency. Add those products together. Then divide by the total frequency.
The mean of 54 is an estimate, not an exact figure. It assumes values sit evenly spread within each class.
e) The median class
The median is the middle value when everything is in order.
Add the frequencies one class at a time until you pass halfway. The class you land in is the median class.
f) Histograms
A histogram shows a grouped distribution as bars.
The bars touch each other. That is what separates a histogram from a bar chart. The classes run continuously, with no gaps between them.
The tallest bar is the modal class, so a histogram shows you the mode at a glance.
Reading the shape tells you more than any single number. Here most learners cluster in the middle, with fewer at the extremes.
g) Where this is used
A school analyses exam performance by grade bands. A county health office groups patient ages to plan services. A shopkeeper groups daily sales to see which range is most common.
Words to know
- Grouped data -- data organised into classes, with only the frequency of each class recorded.
- Class boundaries -- the true dividing values between adjacent classes, lying halfway between their limits.
- Class width -- the difference between the upper and lower boundaries of a class.
- Class midpoint -- the average of the class limits, used to represent all values in that class.
- Modal class -- the class with the highest frequency.
:::checkpoint Check yourself
- What is the frequency of the class 61-80 in the table above?
- Find the midpoint of the class 21-40.
- Which class is the modal class, and why?
- Why do the bars of a histogram touch? :::
Bridge to practice
The exercises begin with class boundaries, widths and midpoints, move through estimating the mean and identifying modal and median classes, and finish with interpreting histograms and comparing distributions. When calculating a mean, set out the midpoints and products in a clear column before dividing; almost every error in this topic is a bookkeeping slip rather than a conceptual one.