Correlations from the above data calculate the covariance

Question

Econ 310, Spring 2014- Week 2:

Question 1: Calculate the mean, median, mode, and variance of the following students grades:

Table: Data Categories
Student	Grade
Tom	80
Sean	90
Ed	60
Ben	70
Nate	80

How do you interpret the standard deviation?  Here's a "Rule of Thumb", dubbed the "Empirical Rule":  If the histogram of the data is approximately bell-shaped, then:

i. Approximately 68% of all observations fall within 1 standard devation of the mean.

ii. Approximately 95% of all observations fall within 2 standard devation of the mean.

iii. Approximately 99.7% of all observations fall within 3 standard devation of the mean.

But,  more  generally, Chebysheff's Theorem says: the fraction of observations in any sample or population that lie within k standard deviations of the mean is at least:

1 - 1/k² for k > 1              

Another way of measuring variation is through percentiles.  For example, the 25^th percentile of data (first quartile) is where 25% of the data is below a value and 75% of the data is above.  The interquartile range is the difference between the third quartile and the first quartile.

Question 2: Calculate the variance of the following grade data. How much data falls within 1 standard deviation of the data? Does it correspond with the empirical rule? Does it correspond with Chebysheff's Theorem? What is the interquartile range?

Table 3: Data Categories
Student	Grade	Did you eat ice cream before the exam?
1	1	0
2	2	0
3	2	1
4	3	1
5	3	1
6	3	0
7	4	0
8	4	1
9	5	1
10	6	0

Question 3: Correlations. From the above data, calculate the covariance and correlation between grade and ice cream consumption. Can you infer anything? What does the correlation number mean?

Question 4: Interpret this figure plot