Question

Which histogram depicts a higher standard? deviation? 0 2 4 6 8 10 Frequency 40 44 48 52 56 60 A histogram has a horizontal axis labeled from 40 to 60 in increments of 4 and a vertical axis labeled “Frequency” from 0 to 10 in increments of 1. Ten vertical bars of class width 2 extend from the horizontal axis and have heights as follows: 40 to 42, 1; 42 to 44, 2; 44 to 46, 5; 46 to 48, 6; 48 to 50, 8; 50 to 52, 8; 52 to 54, 6; 54 to 56, 7; 56 to 58, 3; 58 to 60, 2. 0 2 4 6 8 10 Frequency 30 40 50 60 70 A histogram has a horizontal axis labeled from 30 to 70 plus in increments of 10 and a vertical axis labeled “Frequency” from 0 to 10 in increments of 1. Nine vertical bars of class width 5 extend from the horizontal axis and have heights as follows: 30 to 35, 2; 35 to 40, 2; 40 to 45, 2; 45 to 50, 6; 50 to 55, 9; 55 to 60, 7; 60 to 65, 5; 65 to 70, 4; 70 to 72, 3. ?(a) ?(b) Choose the correct answer below. A. Histogram a depicts the higher standard? deviation, because the bars are higher than the average bar in b. B. Histogram a depicts the higher standard? deviation, because the distribution has more dispersion. C. Histogram b depicts the higher standard? deviation, because the distribution has more dispersion. D. Histogram b depicts the higher standard? deviation, since it is more bell shaped.

Answer 1

The standard devaition of the data values can understand as spread of the data points. Hence larger the dispersion larger will be the standard deviation.

The standard deviation for histogram a is calculated as follow,

Step 1: The data value for histogram a is,

Bin	Frequency
42	1
44	2
46	5
48	6
50	8
52	8
54	6
56	7
58	3
60	2

Step 2: The mean is defined as,

Where N is the total number of data points and summation X is calculated by summing the multiplication of the class and frequency,

Class	Frequency	Class*Frequency
42	1	42
44	2	88
46	5	230
48	6	288
50	8	400
52	8	416
54	6	324
56	7	392
58	3	174
60	2	120
Sum	48	2474

Now the mean is,

Step 3, The standard deviation calculated by,

Class	Frequency	Class*Frequency
42	1	42	91.04347
44	2	88	113.7536
46	5	230	153.5505
48	6	288	75.26056
50	8	400	19.01397
52	8	416	1.680531
54	6	324	36.26032
56	7	392	139.1369
58	3	174	125.1301
60	2	120	143.0867
	48	2474	897.9167

Similarly the standard deviation for histogram b is obtained,

Class	Frequency	Class*Frequency
35	2	70	959.22
40	2	80	571.22
45	2	90	283.22
50	6	300	285.66
55	9	495	32.49
60	7	420	67.27
65	5	325	328.05
70	4	280	686.44
72	3	216	684.03
Sum	40	2276	3897.6

Hence option C. Histogram b depicts the higher standard deviation, because the distribution has more dispersion. is correct