To determine which group has greater variation in terms of age distribution, we'll calculate the standard deviation for each group. The group with the higher standard deviation will have greater variation.

Step 1: Find the Mean (Average) Age

1. Group A: Ages are 18, 19, 20, 21, 31.

   • Mean for Group A:
     $\text{Mean}_A = \frac{18 + 19 + 20 + 21 + 31}{5} = \frac{109}{5} = 21.8$

2. Group B: Ages are 26, 20, 18, 21, 18.

   • Mean for Group B:
     $\text{Mean}_B = \frac{26 + 20 + 18 + 21 + 18}{5} = \frac{103}{5} = 20.6$

Step 2: Calculate the Variance

1. Group A:

   • Variance for Group A:
     $\text{Variance}_A = \frac{(18 - 21.8)^2 + (19 - 21.8)^2 + (20 - 21.8)^2 + (21 - 21.8)^2 + (31 - 21.8)^2}{5}$ $\text{Variance}_A = \frac{14.44 + 7.84 + 3.24 + 0.64 + 84.64}{5} = \frac{110.8}{5} = 22.16$

2. Group B:

   • Variance for Group B:
     $\text{Variance}_B = \frac{(26 - 20.6)^2 + (20 - 20.6)^2 + (18 - 20.6)^2 + (21 - 20.6)^2 + (18 - 20.6)^2}{5}$ $\text{Variance}_B = \frac{29.16 + 0.36 + 6.76 + 0.16 + 6.76}{5} = \frac{43.2}{5} = 8.64$

Step 3: Compare the Variances

Since the variance of Group A (22.16) is greater than the variance of Group B (8.64), Group A has a greater variation in age distribution.

Answer

Group A has greater variation in terms of age distribution.

Would you like to see the calculations for standard deviation as well or have any further questions?

Additional Questions:

1. What would be the standard deviation for each group based on the variances calculated?
2. How would the range compare between Group A and Group B?
3. If you added an age of 35 to Group A, how would it affect the variation?
4. What if we removed the age of 31 from Group A—would Group B then have greater variation?
5. How does the mean impact the variation in each group?

Tip:

Variance measures the spread of the data, but the standard deviation (square root of the variance) gives a more intuitive measure since it's in the same units as the data (years in this case).