01 · Complete bars
Bars over [0,5) and [5,15) have densities 2 and 3. Find their frequencies and total.
Hint
Use widths 5 and 10.
Worked solution
Frequencies 10 and 30; total 40.
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Histogram areas and estimated counts
Recover complete class counts and estimate threshold or interval counts using histogram areas, with explicit uniformity assumptions and appropriate precision.
Before you startFrequency density, areas and fractions of intervals.
01 / Identify the required area
Break an interval into the bars it crosses.
With true density axes, count in a full class = width×density.
For part of a class, selected width×density is a model-based estimate.
A histogram’s flat top does not prove the hidden observations were evenly distributed. The picture represents the average frequency per unit across a class; partial-bar calculations use an additional uniformity assumption.
Count the shaded areaExplore
Classes [0,10), [10,20), [20,40) have frequencies 6, 8, 12. Total n = 26. Density is constant within each drawn bar.
Below 5: estimated count 3 under uniform within-class spread.
Complete class counts are known. A fraction of a bar estimates how its observations divide only when a within-class model is assumed.
02 / Recover complete class counts
Use each bar’s own width.
A bar covers [20,40) with density 0.6.Worked example
Full width = 20
Use the boundaries.
Full class count = 20×0.6 = 12
The count is represented by the whole rectangle.
Two adjacent full bars contribute the sum of their areas
Do not multiply the total width by just one of their heights.
02 · Different heights
Why is (15 − 0)×3 wrong for the total in question 1?
Hint
Density 3 applies only to the second interval.
Worked solution
It assigns the second height to the first interval too. Add 5×2 + 10×3 = 40 instead of 45.
03 / Take a fraction of a class
The fraction of width becomes an estimated fraction of count.
Watch: partial width gives a modelled count
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Estimated partial count = full class frequency × selected width / full width.
Equivalently, multiply the selected width by that class density.
03 · Half a class
Class [10,20) contains 8 observations. Estimate how many are below 15 within that class.
Hint
The selected width is 5 of the full 10.
Worked solution
Estimate 8×5/10 = 4, assuming uniform spread within the class. The actual count need not be four.
04 · A non-integer estimate
Class [0,12) contains 5 observations. Estimate the count in [0,3).
Hint
The selected interval is one quarter of the width.
Worked solution
Estimate 5×3/12 = 1.25. This is a model estimate, not an exact possible observed count; report or round only as the question requires.
04 / Cross several bars
Add complete and partial contributions.
05 · One boundary crossing
Using the model, estimate the count in [5,15).
Hint
Take the upper half of the first bar and lower half of the second.
Worked solution
Estimate 6×5/10 + 8×5/10 = 3 + 4 = 7.
06 · Two partial ends
Using the model, estimate the count in [5,30).
Hint
Include the complete middle class.
Worked solution
Estimate 3 + 8 + 12×10/20 = 17. The first and last contributions assume uniformity; the middle count is known.
05 / Use a complementary count
Check which side includes the threshold.
07 · At least 25
Using the model, estimate how many observations are at least 25.
Hint
Only [25,40) contributes.
Worked solution
Estimate 12×15/20 = 9. Equivalently, estimated count below 25 is 6 + 8 + 3 = 17, so 26 − 17 = 9.
08 · Boundary exactness
How many model observations are at least 20?
Hint
The last class includes 20.
Worked solution
Exactly 12 under the stated [lower,upper) classes. A threshold at a supplied class boundary can use whole class frequencies without within-class interpolation.
06 / Divide by the right total
An estimated count leads to an estimated percentage.
09 · Percentage below 15
Estimate the percentage of model observations below 15.
Hint
The estimated count is 6 + 4 = 10, out of 26.
Worked solution
About 100×10/26 = 38.46%, under the uniformity model. The exact interior-threshold percentage is not determined by the grouped counts alone.
10 · Compare two samples
A selected histogram area represents 8 observations out of 20 in A and 12 out of 40 in B. Which proportion is larger?
Hint
Normalise each count by its own sample size.
Worked solution
A: 40%; B: 30%. B has the larger count but the smaller proportion.
07 / Find a threshold from a target count
Convert the remaining count into a width.
Find the model median by asking for 13 observations below a threshold.Worked example
The first class contributes 6
Another 7 are needed from the middle class.
Middle density = 8/10 = 0.8
A count of 7 corresponds to width 7/0.8 = 8.75.
Estimated median = 10 + 8.75 = 18.75
This agrees with linear cumulative interpolation.
11 · Target below a value
Using the model, estimate t such that 20 observations are below t.
Hint
Fourteen lie below 20, so 6 more are needed from the final class.
Worked solution
Final density 0.6 gives width 6/0.6 = 10. Thus t ≈ 30.
12 · Target too large
Can a threshold contain 30 of the model’s observations below it?
Hint
The total is 26.
Worked solution
No. A cumulative target cannot exceed the whole dataset count. Check the requested count or whether the diagram covers a larger dataset.
08 / Separate possible counts from estimates
Grouping permits many different within-class arrangements.
13 · What is guaranteed?
Class [10,20) contains 8 observations. Without a uniformity assumption, what are the possible counts below 15 within it?
Hint
All eight might lie on either side of 15.
Worked solution
Any integer from 0 to 8 is possible. The estimate 4 is not a guaranteed count.
14 · Mean and SD alone
You know only a dataset’s mean and SD. Can you perform a partial-histogram-area calculation to find the count beyond one SD?
Hint
No histogram frequencies or distribution shape have been given.
Worked solution
No. A mean and SD do not determine the required areas. Additional class counts or an explicitly stated distributional model are needed.
09 / Add the right rectangles
State when a count becomes an estimate.
Recover complete class areas exactly when the scale is known. For thresholds inside classes, state uniformity, combine partial and full contributions and use the correct total for percentages. Do not turn a modelled fractional count into a claim about exact observed records.
Section 1 of 9 · Identify the required area