Sample Mean, Variance, Standard Deviation and Quartiles

Problem 1: Calculating sample mean, sample variance and sample standard deviation

7 8 7 8 8 11 7 9 10 12 9 10 12
10 10 9 8 11 11 11 9 9 12 10 11 10

The table above gives the shoes sizes of 26 members of a football club.

Tasks:
- Enter the data into column A in an Excel sheet. Use the first rows for titles/descriptions of each column.
- In column B calculate the squares of each value.
- Calculate the sum of all x and all x2 and from this calculate the sample mean, sample variance and sample standard deviation once according to SCHAUM and once according to the lecture notes.
- Create a frequency table on the side.
- From the table, following the method above, calculate the sample mean, sample variance and sample standard deviation using the corresponding formula for the sums. The function SUMPRODUCT might be useful here. Make sure they agree with your first value each time.
- Finally use the original table to obtain sample mean, sample variance and sample standard deviation using the Data - Data analysis option selecting Descriptive Statistics and ticking Summary statistics. Use it also to find the 7th smallest and the 5th largest value for this data set.
- Compare your calculated values with those given by Excel.

Questions
- Which formula does Excel use for calculating the sample variance and sample standard deviation?
- What are the median and the mode of this sample?

Problem 2: Identifying potential outliers

- Open the data on hurricanes.
- Using a column chart plot the data
- Using Data ? Data Analysis ? Rank and Percentile identify the lower and upper quartile and the median by interpolating between the corresponding kth statistics. Using the box and whisker method identify potential outliers. (Careful data is given in descending order!)
- Using Data ? Descriptive Statistics ? Summary statistics find mean and standard deviation.
- Use these values to calculate the z-values for all data point. Identify potential outliers again.
Recall the z-value is the number of standard deviations that a value is away from the mean. (Lecture 4)
- Copy the table twice, once deleting the top hurricane and once the top two hurricanes.
- Repeat the summary statistics and the analysis for outliers using z-values.

Questions
- How do sample mean, sample standard deviation and quartiles for the hurricane data change when the extreme value(s) are excluded?
- What do you observe for the recalculated z-values when excluding the extreme case(s)?
- Can you give reason for including/excluding the top two hurricanes?