Data summaries, histograms and interpretation
Diperkenalkan| English | Makna |
|---|---|
| frequency density/ˈfriːkwənsi ˈdensɪti/ | densitas frekuensi |
Can one average tell the whole story?
- Two groups have the same median but different spread. One summary cannot describe both location and consistency.
- This lesson studies frequency density 频率密度: Frequency divided by class width, used as histogram height.
The method
- Compare an appropriate average and spread in context.
- A histogram uses area for frequency, so height=frequency/class width.
- Grouped estimates assume representative values within intervals.
In a histogram with unequal class widths, which bar contains the most observations?
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Histogram area equals frequency. A wide bar of low density can hold more observations than a narrow tall bar; height alone shows density, not frequency.
A worked case
A class from 10 to 20 with frequency 30 has density 30/10=3. A class from 20 to 40 with frequency 20 has density 20/20=1. Its wider bar must not be mistaken for a larger density. For values 2,4,4,6,9, the median is 4 and mean is 5.
Data summaries, histograms and interpretation
Compare an appropriate average and spread in context
Choose a centre and a spread measure for the worked observations.
Find the density for frequency 30 and class width 10.
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Density=frequency/class width=30/10=3.
Where it goes wrong
- The tallest histogram bar need not contain the most observations.
- A grouped mean is an estimate.
- Correlation does not prove causation, and extrapolation extends beyond the observed range.
Not true: A histogram bar's height equals its frequency, even with unequal class widths. With unequal class widths the height is frequency density; frequency is the area of the bar.
Find the mean of 2,4,4,6,9.
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The total is 25 across five observations: mean=25/5=5.
With unequal class widths the height is frequency density.
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True. With unequal class widths the height is frequency density; frequency is the area of the bar.
Applying it
- Choose a display that fits the data type.
- Give both a numerical comparison and what it means for the population; do not infer more precision than the sample supports.
Find the median of 2,4,4,6,9.
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The sorted list is 2,4,4,6,9. Its third observation is 4.
Match each statistic to its definition.
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The mean uses every value, the median resists outliers, and frequency density makes histogram areas carry the frequencies.
In your course
- Applications and Interpretation HL, the course first assessed in 2021.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Frequency density: Frequency divided by class width, used as histogram height.
- Compare an appropriate average and spread in context.
- Watch for: the tallest histogram bar need not contain the most observations.
Write from a mean of 9
- A class of width 2 has frequency 22.
- The values 5, 8, 8, 11 and 13 have a mean and a median. Bar height is not always the frequency.
A class of width 2 has frequency 22, and the values are 5, 8, 8, 11 and 13. What are the frequency density, the mean and the median?
Frequency density is frequency divided by class width. The mean divides the sum by how many values there are.
What is the mean of 5, 8, 8, 11 and 13?
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The values sum to 45, and 45 divided by 5 is 9. Other options — 8 (the median); 11 (the frequency density).