When the Bars Are Different Widths

Mathematics · Data & Statistics · ages 15-16

Name ______________________ Date ____________

On a histogram, the height of a bar gives the frequency of its class.
Circle one: True False
Why do the bars on a histogram touch each other, with no gaps?
- Because a chart with gaps in it looks untidy and hard to read
- Because one class ends exactly where the next begins, leaving no stretch of scale for a gap
- Because every class on a histogram has to be drawn the same width, so the bars line up neatly
- Because the frequencies always add up to 100
A class 10 to 20 has frequency 50. What is the frequency density?
Answer: ______________
A class holds 18 values and is 6 units wide. What is its frequency density?
Answer: ______________
A class 20 to 30 has frequency 25. What is the frequency density?
- 25
- 10
- 250
- 2.5
A class 0 to 20 has frequency 60, so density 3. How many fall between 5 and 10?
- 15
- 3
- 30
- 6
A class holds 42 values and runs from 6 to 20. What is its frequency density?
Answer: ______________
A bar covering 20 ≤ x < 45 is drawn at a frequency density of 0.8. How many values are in that class?
Answer: ______________
A histogram is used to estimate how many values lie between 12 and 14, inside a class running from 10 to 20. What has to be assumed?
- That the class is the widest one on the chart
- That the values inside that class are spread evenly across it
- That no values lie exactly on 12 or on 14, so the slice has clean ends to measure between
- That the frequency density is a whole number
A histogram's classes are all 1 unit wide. What happens to the frequency densities?
- They all become 1
- They equal the frequencies, since dividing by 1 changes nothing
- They cannot be worked out
- They become the class widths, because dividing a frequency by 1 leaves the width behind

Answer key

For grown-ups. Fold this page away before handing over the rest.

When the Bars Are Different Widths W1-mt_0QAOgzD-Qw-s1

False · The height is the frequency density, which is the frequency shared out over the class width. Only when a class happens to be 1 unit wide do the two match.
Because one class ends exactly where the next begins, leaving no stretch of scale for a gap · One class stops at the number the next one starts at, so there is no part of the scale left over. A gap would claim values were impossible there.
5 · Width 10, so 50 over 10 = 5.
3 · Frequency density is frequency divided by class width: 18 / 6 = 3. That is how many values fall in one unit of the scale.
2.5 · 25 over width 10 = 2.5. The 250 option multiplies.
15 · The slice width is 5, and 5 times density 3 = 15. The 30 option wrongly halves the whole class.
3 · The class width is 20 - 6 = 14, so the density is 42 / 14 = 3 values per unit.
20 · The class is 45 - 20 = 25 wide, so the area is 0.8 x 25 = 20 values.
That the values inside that class are spread evenly across it · The bar is drawn at one height across the whole class, which treats every part of it as equally crowded. Taking a slice of the area only works if that is roughly true.
They equal the frequencies, since dividing by 1 changes nothing · Density is frequency divided by width, and dividing by 1 leaves the frequency alone. That is why equal-width charts can plot frequency directly and get away with it.