Question

A survey asked parents of children aged ten and under how many birthday parties they attended last year. Let X represent the number of birthday parties. The probability distribution is given below. Find the mean and the standard deviation of the probability distribution using Excel. Round the mean and standard deviation to three decimal places.

x P(x) 1 0.0303 2 0.0639 3 0.0197 4 0.003 5 0.0164 6 0.0454 7 0.0981 8 0.0648 9 0.0657 10 0.0124 11 0.0118 12 0.0539 13 0.0497 14 0.0648 15 0.0373 16 0.0475 17 0.0224 18 0.0191 19 0.0088 20 0.0406 21 0.0445 22 0.1202 23 0.0597

Answer 1

Solution:
Mean can be calculated as
Mean = Summation(Xi*P(Xi))

X	P(X)	Xi*P(Xi)
1	0.0303	0.0303
2	0.0639	0.1278
3	0.0197	0.0591
4	0.003	0.012
5	0.0164	0.082
6	0.0454	0.2724
7	0.0981	0.6867
8	0.0648	0.5184
9	0.0657	0.5913
10	0.0124	0.124
11	0.0118	0.1298
12	0.0539	0.6468
13	0.0497	0.6461
14	0.0648	0.9072
15	0.0373	0.5595
16	0.0475	0.76
17	0.0224	0.3808
18	0.0191	0.3438
19	0.0088	0.1672
20	0.0406	0.812
21	0.0445	0.9345
22	0.1202	2.6444
23	0.0597	1.3731

Mean = 12.809
Standard deviation = Sqrt(summation((Xi-mean)^2*P(Xi))

X	P(X)	Xi*P(Xi)	(Xi-Mean)	(Xi-mean)^2	(Xi-mean)^2*P(Xi)
1	0.0303	0.0303	-11.809	139.452481	4.2254101743
2	0.0639	0.1278	-10.809	116.834481	7.4657233359
3	0.0197	0.0591	-9.809	96.216481	1.8954646757
4	0.003	0.012	-8.809	77.598481	0.232795443
5	0.0164	0.082	-7.809	60.980481	1.0000798884
6	0.0454	0.2724	-6.809	46.362481	2.1048566374
7	0.0981	0.6867	-5.809	33.744481	3.3103335861
8	0.0648	0.5184	-4.809	23.126481	1.4985959688
9	0.0657	0.5913	-3.809	14.508481	0.9532072017
10	0.0124	0.124	-2.809	7.890481	0.0978419644
11	0.0118	0.1298	-1.809	3.272481	0.0386152758
12	0.0539	0.6468	-0.809	0.654481	0.0352765259
13	0.0497	0.6461	0.191	0.036481	0.0018131057
14	0.0648	0.9072	1.191	1.418481	0.0919175688
15	0.0373	0.5595	2.191	4.800481	0.1790579413
16	0.0475	0.76	3.191	10.182481	0.4836678475
17	0.0224	0.3808	4.191	17.564481	0.3934443744
18	0.0191	0.3438	5.191	26.946481	0.5146777871
19	0.0088	0.1672	6.191	38.328481	0.3372906328
20	0.0406	0.812	7.191	51.710481	2.0994455286
21	0.0445	0.9345	8.191	67.092481	2.9856154045
22	0.1202	2.6444	9.191	84.474481	10.1538326162
23	0.0597	1.3731	10.191	103.856481	6.2002319157

Standard deviation = sqrt(46.299) = 6.804