Frequency summaries and grouped estimates · Foundation
| English | 中文 | Pinyin |
|---|---|---|
| modal class/ˈməʊdl klæs/ | 众数组 | zhòng shù zǔ |
A table stores how often each value occurs. Its mean needs the frequencies as weights; averaging the listed values alone loses the repeated observations.
- A table stores how often each value occurs. Its mean needs the frequencies as weights; averaging the listed values alone loses the repeated observations.
- This lesson studies modal class 众数组: The class interval containing the greatest frequency.
Choose the mathematical structure
- For exact value frequencies, mean is sum(value×frequency)/total frequency. Locate the median using cumulative counts. For interval data, use class midpoints to estimate a mean; the exact values are unknown. The modal class has greatest frequency, which need not be the tallest unequal-width histogram bar. Compare a suitable average and spread, in context, and state the limitations of grouping or outliers.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines modal class?
The class interval containing the greatest frequency.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Values 1,2,3 with frequencies 2,5,3 give total 10 and weighted sum 2+10+9=21, so mean is 2.1. The fifth and sixth observations are both 2, giving median 2 and mode 2. For continuous classes 0≤x<10,10≤x<20,20≤x<30 with frequencies 2,5,3, midpoints 5,15,25 give estimated sum 10+75+75=160 and estimated mean 16. The modal class is 10≤x<20. The range of individual observations cannot be recovered exactly from these intervals. For values 2,4,4,6,24, mean is 8 and median 4; the unusually large value raises the mean. Comparing two groups should describe both a typical value and variation, rather than selecting whichever summary favours a claim.
Frequency summaries and grouped estimates
For exact value frequencies, mean is sum(value×frequency)/total frequency
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find mean of values 1,2,3 with frequencies 2,5,3.
Weighted total 21 divided by count 10.
Test a tempting shortcut
- The grouped mean is an estimate because all members are represented by a midpoint. A median is not found by averaging the class labels. Outliers can alter the mean/range substantially.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A grouped midpoint mean is always the exact mean of the original measurements. This claim is false. Explain which definition or assumption it violates.
Find midpoint-estimated mean for classes 0–10,10–20,20–30 with frequencies 2,5,3.
Weighted midpoint sum 160 divided by 10.
A grouped midpoint mean is always the exact mean of the original measurements.
The grouped mean is an estimate because all members are represented by a midpoint. A median is not found by averaging the class labels. Outliers can alter the mean/range substantially.
Interpret a new situation
- AQA S4/S5 Foundation includes appropriate mean/median/mode/modal class and range, with grouped data and outlier awareness. Higher quartiles/box plots are taught separately.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find median of 2,4,4,6,24.
Sort and select the central third observation.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.6. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The class interval containing the greatest frequency. Choose the relationship, show the method, check its assumptions and interpret the result.