GCSE Histograms Mark Scheme

1.	(a)	60
40	2
B1 cao
B1 cao
(b)	correct bars	2
B1 for 30 < x ≤ 40 with an area of 2½ squares
B1 for 40 < x ≤ 70 with an area of 3 squares
SC:  give M1 if clearly using area or frequency density
[4]
2.	(a)	Frequency densities of 8 ÷ 10 = 0.8
16 ÷ 10 = 1.6, 15 ÷ 5 = 3, 12 ÷ 5 = 2.4
6 ÷ 20 = 0.3	3
B1+ B1 + B1 for each correct column shown on histogram
If B0, then M1 for clear attempt to use frequency density or area
(b)	18, 14, 10, 8	2
	1.8 × 10 = 18, 2.8 × 5 = 14, 2 × 5 = 10, 0.4 × 20 = 8
B2 all correct
B1 2 or 3 correct
5
€ 5 = 
= 2.5 birds
[5]
3.	Reading top to bottom frequencies are 20; 18; 45; 52	2
B2 all correct
(B1 for one frequency correct)
[2]
4.	(a)	40, 60, 56, 32	2
B2 for all frequencies correct
(B1 for any 1 frequency correct)
(b)		3
B1 for Frequency density label or appropriate units
B2 for 4 correct histogram bars sq
(B1 for 2 bars correct)
[5]
5.	(a)	38 × 5, 36 × 17.5, 30 × 32.5, 46 × 50
(= 190, 630, 975, 2300)
	Σfx = 190 + 630 + 975 + 2300 = 4095
Mean Σfx / Σf = 4095/150
27.3	4
M1 for fx with x within intervals (including ends) at least two consistently
M1 (dep) for fx consistently using midpoints
M1 (dep on 1st M) for use of Σfx / Σf
A1 for 27.3 cao
(b)	Frequency density (number of pictures per cm2)
e.g.
Width   0 to 10	height of rectangle 3.8(k)
Width 10 to 25	height of rectangle 2.4(k)
Width 25 to 40	height of rectangle 2(k)
Width 40 to 60	height of rectangle 2.3(k)
Bars with correct heights, widths, label and scaling	3
B2 for 4 rectangles with correct widths and heights
(B1 for 3 rectangles with correct widths and heights)
B1 for correct label or key and consistent scaling
(SC if 0/3 award M1 if clearly using area or freq. density)
[7]
6.	
F	4	10	24	20	6	
Fd	0.8	1	1.6	2	1.2	
	or
F	4	10	24	20	6	
Fd	4	5	8	10	6	
	Correct histogram	4
M1 use of frequency density as frequency ÷ width (can be implied by two correct frequency densities or two correct bars with different widths) or area (can be implied by one correct bar) to represent frequency
A2 for all 5 histogram bars correct ±½ square
(A1 at least 3 correct histogram bars ±½ square)
A1 for correct label and scale numbered appropriately or for key and consistent scaling

	Examples
		
[4] 
7.	(a)	25
	16	2
M1 for correct use of frequency density to find a unit of area (for example 1cm2 = 2.5 or 1 small square = 0.1) or the area of one block.
A1 cao
(b)	Correct black (1 cm high between 40 and 60)	1
B1 for correct black
[3]
8.	(a)	Freq = FD × int width = 0.018 × 1000 = 18
Or = 18 × 1 = 18
	= 0 .010 × 2000 = 20 or 10 × 2 = 20
	= 0.006 × 2000 = 12 or 6 × 2 = 12
OR
No of small squares = 200     Total freq = 16
	So 1 small square = 16 ÷ 200 = 0.08
	9 × 25 × 0.08 = 18
	10 × 25 × 0.08 = 20
	6 × 25 × 0.08 = 12
OR 8 cm2 = 16 so 1 cm2 = 2 etc
18, 20, 12		2
M1 use of Freq = FD × int width or attempt to find freq of 1 standard square
(or one answer correct)
A1 cao: all three
(b)	FD = Freq ÷ int width = 16 ÷ 2000 = 0. 008 so 4 sqs up
4000-6000
4 cm high
	= 8 ÷ 4000 = 0.002 so 1 sq up
OR
16 ÷ 0.08 = 200 200 ÷ 25 = 8 so 4 sqs up
8 ÷ 0.08 = 100 100 ÷ 25 = 4 so 1 square up
OR 16 ÷ 2 = 8 so 4 sqs up etc
8000-12000
1 cm high	2
B1 4000-6000; 4 cm high
B1 8000-12000; 1 cm high
or
if B0, M1 use of Freq = FD × int width or attempt to find freq of 1 standard square
[4]
9.	(a)	8, 6	2
B1 cao
B1 cao
(b)	Bars of ht 4cm, 5cm	2
B1 for height = 4 ± 1 mm
B1 for height = 5 ± 1 mm
[4]
10.	(a)	24
= 30	2
B1 24 cao
B1 30 cao
 
(b)	Column 60 ≤ w < 75 at 10
Column 75 ≤ w < 95 at 4
10
= 4	2
B1 10 and correct width (tol ±  small square)
B1 4 and correct width (tol  small square)
[4]
11.	(a)	390	2
B1cao
	400
B1 cao
(b)	Correct bar	1
B1 for correct bar
[3]
12.	(a)	Heights 24, 32	2
B1 cao for bar from 15 – 17.5, height 24 × 2mm squares
B1 cao for bar from 17.5 – 20, height 32 × 2mm square
(b)	Freqs 40, 20, 15	2
B2 cao for all 3 correct
(B1 for any 1 or 2 correct)
(c)	Area up to 12.5 = 220x
Area above 21 = 156x
Frequency 
78	3
M1 for attempt to find area up to 12.5 and area above 21 consistently
M1 for  or  or  oe
A1 78 cao
SC: If no marks earned B1 for 2mm2 = 1 person oe
[7]