Statistics Review - Mathematics and Statistics for the Environment - page 111

where X i is the observed value of the i ith unit in the sample, n is the number of units in the sample,
and

n

∑

X

1

i =

means to sum up all n of the X values in the sample.
If there are N units in the population, the total of the X values over all units in the population
would be estimated by

ˆ TNX

=

The circumflex (^) over the T is frequently used to indicate an estimated value as opposed to the
true but unknown population value. It should be noted that this estimate of the mean is used for a
simple random sample. It may not be appropriate if the units included in the sample are not selected
entirely at random.

7.3.2 m edian
The median is the value of the central item when the data are arrayed in size.

7.3.3 m ode
The mode is the observation that occurs with the greatest frequency and thus is the most “fashion-
able” value.

7.3.4 r ange
The range is the difference between the values of the highest and lowest terms.

■ EX AMPLE 7.1
Problem: Given the following laboratory results for the measurement of dissolved oxygen (DO) in
water, find the mean, mode, median, and range.

6.5 mg/L, 6.4 mg/L, 7.0 mg/L, 6.9 mg/L, 7.0 mg/L

Solution: To find the mean:

n

∑ 1

X

i

6.5 mg/L + 6.4 mg/L + 7.0 mg/L + 6.0 mg/L + 7.0 mg/L

i =

X = = = 6.58 mg/L

n 5

The mode is 7.0 mg/L, the number that appears most often.

Now arrange the measurements in order:

6.4 mg/L, 6.5 mg/L, 6.9 mg/L, 7.0 mg/L, 7.0 mg/L.

The median is 6.9 mg/L, the central value, and the range is 0.6 mg/L (7.0 mg/L - 6.4 mg/L).