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IG.M20 Histograms and Frequency Density

EDEXCEL 4MA1 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A grouped frequency table records heights of 50 students in cm: 140≤h<150: 5, 150≤h<160: 12, 160≤h<170: 18, 170≤h<180: 10, 180≤h<190: 5. Calculate the frequency density for the class 150≤h<160.

(Total for Question 1 is 1 mark)

From the same grouped table as Question 1, calculate the frequency density for the class 160≤h<170.

(Total for Question 2 is 1 mark)

A class 120≤x<140 has frequency 6 and a class width 20. Calculate its frequency density and give units used to describe the vertical axis on a histogram of x.

(Total for Question 3 is 2 marks)

A histogram will be drawn for the distribution: class 0≤t<5 freq 8, 5≤t<15 freq 20, 15≤t<20 freq 12. Calculate the frequency density for 5≤t<15 (note class width is not 10 for every class).

(Total for Question 4 is 2 marks)

A grouped table for speeds (km/h) shows: 0≤s<20: frequency 3 (width 20), 20≤s<40: frequency 12 (width 20), 40≤s<60: frequency 25 (width 20). What height should the bar for 40≤s<60 have on a histogram where vertical axis is frequency density?

(Total for Question 5 is 2 marks)

A student draws a histogram for daily rainfall (mm) with unequal widths. The class 0≤r<2 has area 4 on the histogram. The class 2≤r<5 has frequency 9 and width 3. Calculate the frequency density for 2≤r<5 and then the height of the bar. Show your working.

(Total for Question 6 is 3 marks)

A histogram shows the following bars (frequency densities shown): class 10≤x<20 height 0.5, class 20≤x<30 height 1.0, class 30≤x<50 height 0.6. Calculate the frequency in the class 30≤x<50 by using area of the bar.

(Total for Question 7 is 3 marks)

Complete the following: a histogram for exam marks has bars for classes 0≤m<10, 10≤m<30, 30≤m<50. The class 10≤m<30 has frequency 40 and width 20. The histogram shows the height of this bar as 2.0. Using this, calculate the total number of students represented by the three classes if the other bars have heights 1.5 (width 10) and 1.8 (width 20) respectively. Show working for each class area.

(Total for Question 8 is 4 marks)

A histogram bar for the interval 50≤y<70 has height 0.9. Estimate the frequency in the sub-interval 60≤y<70 (the upper half of that class) assuming uniform distribution within the class. State your method.

(Total for Question 9 is 3 marks)

The histogram below (data described) has a bar 0≤z<4 with height 1.5, a bar 4≤z<7 with height 2.4, and a bar 7≤z<10 with height 1.2. Use these to complete the frequency column of the table: classes 0-4, 4-7, 7-10. Give each frequency as an integer.

(Total for Question 10 is 3 marks)

A histogram is given for weights of parcels with unequal class widths. The class 0≤w<2 has frequency 8, 2≤w<5 has unknown frequency f, and 5≤w<10 has frequency 30. The histogram shows heights: 0≤w<2 height 4, 2≤w<5 height 3, 5≤w<10 height 6. Complete the frequency f using areas and show steps.

(Total for Question 11 is 4 marks)

A histogram representing age groups has classes 0≤a<10, 10≤a<30, 30≤a<60. The heights are 0.8, 0.6 and 0.4 respectively. Estimate the total population represented by the histogram and explain briefly why heights decrease as class width increases in some histograms.

(Total for Question 12 is 4 marks)

A histogram shows a bar for 100≤x<140 with height 0.75. A researcher asks for the estimated number in 110≤x<120. Explain how to estimate this number and calculate it, assuming uniform distribution within the class.

(Total for Question 13 is 4 marks)

A frequency table is partially blank. Classes: 0≤v<5 width 5 density 0.6, 5≤v<12 width 7 density unknown with frequency 21, 12≤v<20 width 8 density 1.0. Complete the missing density for 5≤v<12 and the total frequency for all classes.

(Total for Question 14 is 3 marks)

Explain briefly why, when drawing a histogram from a grouped frequency table with unequal class widths, the vertical axis must show frequency density rather than frequency. Give a short numerical example to support your explanation.

(Total for Question 15 is 3 marks)

Mark scheme · IG.M20 Histograms and Frequency Density

Question 1

B1 frequency density = frequency / class width = 12 / 10 = 1.2 cao
Answer: 1.2

Question 2

B1 frequency density = 18 / 10 = 1.8 cao
Answer: 1.8

Question 3

M1 finds frequency density = 6 / 20 or writes correct division
A1 frequency density = 0.3 (per unit on x) or 0.3 per cm, cao
Answer: 0.3, units: frequency density in 'frequency per unit x' (e.g. per cm)

Question 4

M1 identifies correct class width = 10 and divides 20 by 10
A1 frequency density = 2.0 cao
Answer: 2.0

Question 5

M1 divides 25 by class width 20
A1 height = 1.25 (frequency per km/h) cao
Answer: 1.25

Question 6

M1 uses density = frequency / width: density = 9 / 3
A1 frequency density = 3.0 cao
A1 height = 3.0 (since height equals frequency density) cao
Answer: frequency density = 3.0, height = 3.0

Question 7

M1 uses area = height * width with width = 50 - 30 = 20
M1 calculates area = 0.6 * 20 = 12
A1 frequency = 12 (people/items) cao
Answer: 12

Question 8

M1 calculates frequency for first class from area: 1.5 * 10 = 15
M1 uses given for middle class: 2.0 * 20 = 40
M1 calculates frequency for third class: 1.8 * 20 = 36
A1 total = 15 + 40 + 36 = 91 cao
Answer: frequencies: 15, 40, 36. Total = 91

Question 9

M1 uses area method and recognises width of full class = 20 and sub-interval width = 10
M1 calculates full class frequency = 0.9 * 20 = 18, then takes half for 10 width = 9
A1 estimate = 9 cao
Answer: 9

Question 10

M1 uses area = height * width for each class
M1 calculates areas: 1.5*4 = 6; 2.4*3 = 7.2; 1.2*3 = 3.6
A1 gives frequencies 6, 7.2 (accept 7.2) and 3.6 (accept 3.6) and if integers required rounding to nearest whole numbers 6, 7, 4 cao
Answer: frequencies (exact): 6, 7.2, 3.6. As integers (nearest): 6, 7, 4

Question 11

M1 uses area = height * width to find frequencies for known bars: first and third
M1 calculates first area = 4 * 2 = 8 and third gives 6 * 5 = 30
M1 sets f = height * width for middle class: f = 3 * (5 - 2) = 3 * 3
A1 f = 9 cao
Answer: f = 9

Question 12

M1 calculates areas: 0.8*10 = 8, 0.6*20 = 12, 0.4*30 = 12
M1 adds correctly to give total 32
M1 gives a correct reason linking height to frequency density and class width, e.g. density = frequency/width so wider classes often have lower heights for same frequency, or to keep area equal to frequency
A1 total = 32 and sensible explanation about density and width, cao
Answer: Total = 32, explanation: heights are frequency densities; density = frequency/width so if class width increases for similar frequencies the height falls to keep area equal to frequency

Question 13

M1 states method: find full class frequency by area then take proportion corresponding to sub-interval width
M1 calculates full class width = 40 and full frequency = 0.75 * 40 = 30
M1 calculates sub-interval width = 10 and proportional frequency = 30 * (10/40) = 7.5
A1 gives estimate 7.5 and comment that this may be rounded or reported as 7 or 8, cao
Answer: Estimate = 7.5 (about 8 people if rounded)

Question 14

M1 uses density = frequency / width to find density for middle class: 21 / 7
M1 calculates densities and frequencies for other classes: first freq = 0.6*5 = 3; third freq = 1.0*8 = 8
A1 density for 5≤v<12 = 3.0 and total frequency = 3 + 21 + 8 = 32 cao
Answer: middle density = 3.0, total frequency = 32

Question 15

M1 states that area of bar must represent frequency, so height must be frequency/width when widths vary
M1 gives a correct short numerical example, e.g. class A width 2 freq 6 density 3, class B width 6 freq 18 density 3 so equal densities give proportional areas
A1 concludes that using frequency as height would misrepresent data when widths differ, cao
Answer: Because area represents frequency. Example: class width 2 freq 6 gives density 3, width 6 freq 18 gives density 3; plotting densities gives areas 2*3=6 and 6*3=18. If heights were frequencies, bars would be 6 and 18 and areas would be 12 and 108, misrepresenting counts.

Histograms and Frequency Density - Worksheets, Questions and Revision