Descriptive Statistics: A Example with Dice
As an example of descriptive statistics we can look at the process of rolling dice. Here we will roll the die twelve times and record the numbers that we got on each roll. The process is repeated seven more times so that we have eight samples of numbers.
Numbers Rolled
|
sample1 |
sample2 |
sample3 |
sample4 |
sample5 |
sample6 |
sample7 |
sample8 |
|
4 |
6 |
3 |
4 |
6 |
5 |
4 |
4 |
|
2 |
5 |
4 |
3 |
5 |
5 |
6 |
4 |
|
5 |
2 |
1 |
6 |
4 |
2 |
2 |
5 |
|
4 |
4 |
3 |
4 |
6 |
1 |
3 |
3 |
|
3 |
2 |
4 |
6 |
2 |
2 |
2 |
3 |
|
1 |
5 |
6 |
6 |
3 |
5 |
6 |
5 |
|
6 |
1 |
3 |
4 |
1 |
2 |
2 |
3 |
|
3 |
1 |
3 |
3 |
5 |
6 |
1 |
2 |
|
4 |
5 |
4 |
2 |
4 |
1 |
4 |
5 |
|
2 |
2 |
4 |
3 |
4 |
5 |
1 |
6 |
|
1 |
3 |
2 |
1 |
5 |
5 |
2 |
2 |
|
2 |
6 |
1 |
4 |
6 |
1 |
2 |
3 |
After selecting Data Analysis from the Tool menu you should select descriptive statistics. Once you complete the dialogue box by inputting the location of the numbers and the upper left hand cell of the location where you will put the results you will get a table that looks similar to what you see below. The average, what we will call the mean, ranges from 2.92 and 4.25, while the median ranges from 2 to 4.5. The mean for the entire 96 observations is 3.48
Descriptive Statistics: Small Samples
|
sample1 |
sample2 |
sample3 |
sample4 |
sample5 |
sample6 |
sample7 |
sample8 |
|
|
Mean |
3.08 |
3.50 |
3.17 |
3.83 |
4.25 |
3.33 |
2.92 |
3.75 |
|
Standard Error |
0.45 |
0.54 |
0.41 |
0.46 |
0.46 |
0.57 |
0.50 |
0.37 |
|
Median |
3 |
3.5 |
3 |
4 |
4.5 |
3.5 |
2 |
3.5 |
|
Mode |
4 |
5 |
3 |
4 |
6 |
5 |
2 |
3 |
|
Standard Deviation |
1.56 |
1.88 |
1.40 |
1.59 |
1.60 |
1.97 |
1.73 |
1.29 |
|
Sample Variance |
2.45 |
3.55 |
1.97 |
2.52 |
2.57 |
3.88 |
2.99 |
1.66 |
|
Kurtosis |
-0.62 |
-1.69 |
0.53 |
-0.46 |
0.00 |
-2.07 |
-0.27 |
-1.00 |
|
Skewness |
0.35 |
0.00 |
0.12 |
0.00 |
-0.81 |
-0.04 |
0.91 |
0.25 |
|
Range |
5 |
5 |
5 |
5 |
5 |
5 |
5 |
4 |
|
Minimum |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
2 |
|
Maximum |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
6 |
|
Sum |
37 |
42 |
38 |
46 |
51 |
40 |
35 |
45 |
|
Count |
12 |
12 |
12 |
12 |
12 |
12 |
12 |
12 |
A "picture" of the results can be generated by following the same procedure except asking for Histogram rather than Descriptive Statistics. Once you have selected the numbers as the input and the location for the output, you can create a histogram. The histograms for the data from the first sample and the combined sample of 96 appear below. In the first histogram it is easy to see that two "3s" were rolled and three "4s," while in the second histogram it is clear that the distribution is beginning to lose its high and low points - something we will discuss later.
