Question

Central Tendency and Variation Activity

1. Roll 4 dice 40 times, recording the sum for each roll.   Do it again so you have two groups of 40 numbers

2. Calculate the mean, median, mode, and midrange for both sets of data. The four numbers should be close to one another for each data set. Write a sentence or two accounting for any discrepancies.

3. Calculate a 5 number summary for both sets of data (it may help to create a tally chart or stem and leaf plot for both sets of data. Write a sentence or two describing any differences between the two sets of number and offer reasons for the discrepancy.

4. Calculate the standard deviation for both sets of data. Which data set is more spread out? Why is this so?       

Answer 1

1.

First data set

We roll 4 dice 40 times and obtain sum of rolls as follows.

18,    8, 14, 16, 15, 10, 14, 14,    5, 15, 16, 14,    4, 15, 16, 18, 19, 14, 16, 17,

15, 16, 9,    17, 23, 18, 14, 12, 10, 14, 22, 14, 21, 12, 14,    8, 15,    8, 10, 20.

Second data set

We roll 4 dice 40 times and obtain sum of rolls as follows.

16, 17, 16,    9, 17, 20, 18,    7, 14, 14,    4, 16, 15, 10, 16, 18, 19, 14, 15, 10,

15, 14, 15, 20, 14, 22, 15, 14, 12, 12, 14, 8, 14, 23, 18, 15, 8,    11, 14,    5

2.

First data set

Mean = 14.25

Arranging the values in increasing order we get,

4, 5, 8, 8, 8, 9, 10, 10, 10, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14,
15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 19, 20, 21, 22, 23.

Median = Average of 20 th and 21 st observations = (14+15)/2 = 14.5

We observe that 14 occurs 9 times, which is highest. So, mode = 14

Midrange = (Maximum value + Minimum value) / 2 = (4+23) / 2 = 13.5

Second data set

Mean = 14.2

Arranging the values in increasing order we get,

4, 5, 7, 8, 8, 9, 10, 10, 11, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14,
15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 19, 20, 20, 22, 23.

Median = Average of 20 th and 21 st observations = (14+15)/2 = 14.5

We observe that 14 occurs 9 times, which is highest. So, mode = 14

Midrange = (Maximum value + Minimum value) / 2 = (4+23) / 2 = 13.5

3.

First data set

Mean = 14.25

Median = 14.5

Mode = 14

Range = 23-4 = 17

Standard deviation = 4.223446

Second data set

Mean = 14.2

Median = 14.5

Mode = 14

Range = 23-4 = 17

Standard deviation = 4.196427

Conclusion-

We obtained these values through practical experiments. So these values have little differences, although these differences are not so much significant.

4.

For first data set, standard deviation = 4.223446

For second data set, standard deviation = 4.196427

Clearly, first data set having higher standard deviation is more spread out. Both of these data sets are expected to more more or less normally distributed and in such instances mean, median and mode are expected to be almost equal. The little differences which are observed are purely based on randomness of our experiment.