Expectation, Variance & Covariance

Summarize a distribution with its expected value and spread, then describe how two variables move together.

By the end of this lesson

You will be able to

Calculate an empirical mean and variance.
Explain expected value as a probability-weighted average.
Interpret the sign of covariance.
Distinguish covariance from standardized correlation.

Expectation and variance summarize one variable

For a discrete random variable, expected value is the probability-weighted average of its possible values. For observed data, the sample mean is the average of the observed values. Variance is the average squared distance from the mean under the chosen population or sample convention; standard deviation is its square root and returns to the original units. Variance grows quadratically when values are rescaled.

values = np.array([2.0, 4.0, 4.0, 6.0])
print(values.mean())
print(values.var())
print(values.std())

Covariance and correlation describe pairs

Covariance is positive when two variables tend to be above or below their own means together, and negative when one tends to be high while the other is low. Its magnitude depends on the units. Pearson correlation standardizes the relationship to a value from -1 to 1 and describes linear association. Neither statistic alone establishes that one variable causes the other.

temperature = np.array([10, 12, 15, 18, 20], dtype=float)
demand = np.array([30, 34, 41, 45, 48], dtype=float)

print(np.cov(temperature, demand, ddof=1)[0, 1])
print(np.corrcoef(temperature, demand)[0, 1])

Apply what you learned

Exercise

Summarize a pair of variables

Calculate descriptive statistics for a small set of observations and describe what the numbers do and do not establish.

Calculate mean, variance and standard deviation for [2, 4, 4, 6].
Use NumPy with an explicit ddof choice and state whether you are using population or sample variance.
Calculate covariance and Pearson correlation for two short arrays.
Multiply one array by 10 and recalculate covariance and correlation.
Explain why correlation is unchanged by a positive rescaling while covariance changes.

Done when

Your output states the variance convention, correlation stays within [-1, 1], and your interpretation separates association from causation.

Free references

Study resources

3 links

01
Calculating the Mean, Variance and Standard Deviation Video
Watch the focused calculation and compare the definitions with the values in your exercise.

Done
02
Covariance, Clearly Explained Video
Watch the single covariance explanation and focus on the sign and unit dependence.

Done
03
Pearson’s Correlation, Clearly Explained Video
Watch the individual correlation explanation and note the difference from covariance.

Done

Check your understanding

Self-test

2 questions

Question 1

What does variance measure?
Question 2

What does Pearson correlation add that covariance does not?

Completion check

Move on when

You can calculate the basic summaries and explain how mean, variance, covariance and correlation differ.

Mark passed and continue Back to module