Construct two different datasets such that the range of the first set is greater than the range of the second set, but the standard deviation of first set is less than the standard deviation of the second set. Can you say that one of your datasets has the smaller variation of the two?
1, 1000, 40, 41, 39,40,40,39,41
31,49,74,26,54,36,55,19
Try those.
@bobpursley thank you very much. I tried those data sets but it didnot work :(
Try 75, 25, 50, 51, 52, 49, 48, 50, 50, 49 for your first equation and
61, 39, 62, 38, 63, 37, 64, 36, 65, 35 for the second.
I have made the distributions with mean = 50 and n = 10 to make the calculations easier for you.
The range only relies on the two most extreme scores, while the SD includes the variation from the mean of all scores.
To construct two different datasets that meet the criteria, we need to consider how the range and standard deviation are calculated and how they are related.
Range: The range of a dataset is the difference between the maximum and minimum values. A larger range means there is a greater spread between the highest and lowest values.
Standard deviation: The standard deviation measures the dispersion or spread of a dataset around the mean. A larger standard deviation indicates that the values are more spread out from the mean, reflecting greater variation.
To satisfy the given conditions, we can have Dataset A and Dataset B as examples:
Dataset A:
Values: 2, 4, 6, 6, 8, 10
Range = Maximum value (10) - Minimum value (2) = 8
Standard deviation = 2.83
Dataset B:
Values: 2, 4, 4, 4, 8, 10
Range = Maximum value (10) - Minimum value (2) = 8
Standard deviation = 2.49
In this example, both datasets have the same range of 8. However, the standard deviation of Dataset A (2.83) is larger than that of Dataset B (2.49), indicating greater variation in Dataset A.
So, even though the range of Dataset A is greater than the range of Dataset B, we cannot say that Dataset A has smaller variation because its standard deviation is higher. Standard deviation takes into account the spread of values around the mean, not just the range.