Two families with 6 members are randomly selected. Each of the fami members took an IQ test. Their scores are as follows.
Family A: 109, 113, 117, 120, 128, 111
Family B: 120, 118, 115, 116, 119, 118
Which of the two families has more consistent scores?
To determine which family has more consistent scores, we need to look at the range and standard deviation of each family's scores.
Family A: Range = 128 - 109 = 19; Standard deviation = 7.7
Family B: Range = 120 - 115 = 5; Standard deviation = 1.4
Since Family B has a smaller range and lower standard deviation, their scores are more consistent. Therefore, Family B has more consistent scores.
To determine which of the two families has more consistent scores, we can calculate the standard deviation for each family's IQ scores. The family with a lower standard deviation will have more consistent scores.
First, let's calculate the standard deviation for Family A:
1. Find the mean of the scores: (109+113+117+120+128+111)/6 = 598/6 = 99.67
2. Subtract the mean from each score and square the result:
(109-99.67)^2 = 86.45
(113-99.67)^2 = 178.56
(117-99.67)^2 = 317.59
(120-99.67)^2 = 417.71
(128-99.67)^2 = 823.45
(111-99.67)^2 = 130.11
3. Find the mean of the squared differences:
(86.45+178.56+317.59+417.71+823.45+130.11)/6 = 1953.87/6 = 325.64
4. Take the square root of the mean squared differences to get the standard deviation:
√(325.64) ≈ 18.04
Next, let's calculate the standard deviation for Family B:
1. Find the mean of the scores: (120+118+115+116+119+118)/6 = 706/6 = 117.67
2. Subtract the mean from each score and square the result:
(120-117.67)^2 = 5.41
(118-117.67)^2 = 0.11
(115-117.67)^2 = 7.11
(116-117.67)^2 = 2.77
(119-117.67)^2 = 1.81
(118-117.67)^2 = 0.11
3. Find the mean of the squared differences:
(5.41+0.11+7.11+2.77+1.81+0.11)/6 = 17.32/6 = 2.89
4. Take the square root of the mean squared differences to get the standard deviation:
√(2.89) ≈ 1.70
Comparing the standard deviations, we find that Family A has a standard deviation of approximately 18.04, while Family B has a standard deviation of approximately 1.70. Since the standard deviation for Family B is lower, it indicates that their scores are more consistent compared to Family A. Therefore, Family B has more consistent scores.