Properties of the Mean

mean definition

The average or mean of a collection of numbers is the sum of all the elements of the collection, divided by the number of elements in the collection.

The methods np.average and np.mean return the mean of an array.

Basic Properties

• It need not be an element of the collection.

• It need not be an integer even if all the elements of the collection are integers.

• It is somewhere between the smallest and largest values in the collection.

• It need not be halfway between the two extremes; it is not in general true that half the elements in a collection are above the mean.

• If the collection consists of values of a variable measured in specified units, then the mean has the same units too.

The mean is a “Smoother”

You can think of taking the mean as an “equalizing” or “smoothing” operation. For example, imagine the entries in not_symmetric above as the dollars in the pockets of four different people. To get the mean, you first put all of the money into one big pot and then divide it evenly among the four people. They had started out with different amounts of money in their pockets ($2, $3, $3, and $9), but now each person has $4.25, the mean amount.

Proportions are Means

If a collection consists only of ones and zeroes, then the sum of the collection is the number of ones in it, and the mean of the collection is the proportion of ones.

zero_one = make_array(1, 1, 1, 0)
sum(zero_one)

0.75

The Mean and the Histogram

The mean of the collection {2,3,3,9}, which I not “halfway point’ of the data. So then what does the mean measure?

to see this, notice that the mean can ce calculated in differnt ways

= 2+3+3+ 9/ 4

When we calculate the mean, each distinct value in the collection is weighted by the proportion of times it appears in the collection

= 2 × 0.25 + 3 × 0.50 + 9 × 0.25

Important consequence of the mean

The mean of a collection depends only on the distinct values and their proportions, not on the number of elements in the collection.

The mean of a collection depends on the distribution of values in the collection

In general, for symmetric distributions, the mean and the median are equal

In general, if the histogram has a tail on the one side (the formal term is “skewed”), then the mean is pulled away from the median in the direction of tail

Example":

this histogram is skewed to the right; it has a right-hand tail

The mean gets pulled away from the medical in the direction of the tail. So we expect the mean compensation to be larger than the median, and that is indeed the case

compensation = sf20215.column(‘Total Compensation’)

percentile(50,compensation)

output: 110305.79

np.mean(compensation)

output: 114725.98

Notes

Distribution of incomes of large populations tend to be right skewed. When the bulk of the population has middle to low incomes, but a very small proportion has high incomes, the histogram has a long, thin tail to the right

The mean income is affected by the tail: the further the tail stretches to the right, the larger the mean becomes. But the median is not affected by values at the extremes of the distribution.