Week 2 Descriptive Statistics and Probability

386311. Our weekly discussion scenario about the requested dollar amount for meals includes a spreadsheet full of numbers. When you are asked to calculate the mean of those numbers, are you calculating a population mean or a sample mean. How do you know? What does the scenario tell you?

2. Given these numbers
14, 22, 555, 6, 8, 14, 6, 14, 14, 5, 4, 25, 12, 6
In Excel:
Calculate the mean:
Calculate the median:
Calculate the mode:

a) Why was the mean so much higher than the median or mode?
b) Why wouldn’t the mean always be the best number to represent the central tendency of a group of numbers?

2. Calculate the range, variance, and standard deviation in EXCEL for the SAMPLE below. Then, calculate the range, variance and standard deviation in EXCEL as if the group of numbers was a POPULATION. Upload your spreadsheet to your submission with the calculations from #2 and #3.
14, 22, 555, 6, 8, 14, 6, 14, 14, 5, 4, 25, 12, 6
Range of the sample:
Range of the population:
Variance of the sample:
Variance of the population:
Standard deviation of the sample:
Standard deviation of the population:

Why did the standard deviation and variance change when you calculated it for the sample and the population for this set of numbers? What about the formula causes this change? (Analyze the formulas – what is different?)

Hint: Review this before you answer 3 and 4.
https://www.mathsisfun.com/data/standard-deviation.html

3. Both variance and standard deviation measure variation in a set of numbers. When calculating variance, we square the distance each value is from the mean. For example, if the mean is 15 and the data point is 11, we take the distance the data point is from the mean (15-11 = 4) and then we square the distance (4 x 4 = 16). However, 11 and 15 are not 16 units apart. So, why are we squaring the distance? Doesn’t that just make the distance look larger than it really is?

4. Standard deviation is the square root of the variance, and both are measures of variation. In #3, we saw that when calculating variance, we square the distance each point is from the mean. Why would you square just to take the square root? Doesn’t that get us back where we started? For example, if we square 4, we get 16, and if we square root 16, we are right back to 4.

5. Explain the following in your own words:
percentile,
the first quartile,
the third quartile,
and the IQR (interquartile range).
Use the list of numbers: 14, 22, 555, 6, 8, 14, 6, 14, 14, 5, 4, 25, 12, 6

to find these values for the data set. Use the Excel commands that are inclusive in this exercise. (=quartile.inc)
6. Find the formula that uses the IQR to determine if a number is an outlier. Use the formula to mathematically show that there is an outlier in the data set in #2.
Data set: 14, 22, 555, 6, 8, 14, 6, 14, 14, 5, 4, 25, 12, 6

7. In #2, use the data for the SAMPLE. Assume it is normally distributed, even if it is not. Use the empirical rule to calculate the range of values that are 1 standard deviation from the mean. Show your work. Do the same for two and three standard deviations from the mean.