Question

A study randomly selected 100 samples, each of which consisted of 100 people, and recorded the number of left-handed people, X. The table below shows the probability distribution of the data. Find the mean and the standard deviation of the probability distribution using Excel.

• Round the mean and standard deviation to two decimal places.

x	P(x)
1	0.01
2	0.01
3	0.04
4	0.02
5	0.02
6	0.09
7	0.08
8	0.04
9	0.07
10	0.02
11	0.04
12	0.04
13	0.16
14	0.08
15	0.08
16	0.02
17	0.02
18	0.08
19	0.04
20	0.04

Answer 1

Solution:
Here we will use discrete probability distribution because total probability of events is 1
So Mean and standard deviation can be calculated as
Mean = Xi*P(Xi)

X	P(X)	(Xi*P(Xi))
1	0.01	0.01
2	0.01	0.02
3	0.04	0.12
4	0.02	0.08
5	0.02	0.1
6	0.09	0.54
7	0.08	0.56
8	0.04	0.32
9	0.07	0.63
10	0.02	0.2
11	0.04	0.44
12	0.04	0.48
13	0.16	2.08
14	0.08	1.12
15	0.08	1.2
16	0.02	0.32
17	0.02	0.34
18	0.08	1.44
19	0.04	0.76
20	0.04	0.8

Mean = (0.01+0.02+0.12+0.08+0.1+0.54+0.56+0.32+0.63+0.2+0.44+0.48+2.08+1.12+1.2+0.32+0.34+1.44+0.76+0.8) = 11.56
Standard deviation can be calculated as
Standard deviation = sqrt((Xi-mean)^2 *P(Xi))

X	P(X)	(Xi*P(Xi))	(Xi-mean)	(Xi-mean)^2	(Xi-mean)^2*P(Xi)
1	0.01	0.01	-10.56	111.5136	1.115136
2	0.01	0.02	-9.56	91.3936	0.913936
3	0.04	0.12	-8.56	73.2736	2.930944
4	0.02	0.08	-7.56	57.1536	1.143072
5	0.02	0.1	-6.56	43.0336	0.860672
6	0.09	0.54	-5.56	30.9136	2.782224
7	0.08	0.56	-4.56	20.7936	1.663488
8	0.04	0.32	-3.56	12.6736	0.506944
9	0.07	0.63	-2.56	6.5536	0.458752
10	0.02	0.2	-1.56	2.4336	0.048672
11	0.04	0.44	-0.56	0.3136	0.012544
12	0.04	0.48	0.44	0.1936	0.007744
13	0.16	2.08	1.44	2.0736	0.331776
14	0.08	1.12	2.44	5.9536	0.476288
15	0.08	1.2	3.44	11.8336	0.946688
16	0.02	0.32	4.44	19.7136	0.394272
17	0.02	0.34	5.44	29.5936	0.591872
18	0.08	1.44	6.44	41.4736	3.317888
19	0.04	0.76	7.44	55.3536	2.214144
20	0.04	0.8	8.44	71.2336	2.849344

Standard deviation = Sqrt(23.5664) = 4.85