Two summary statistics that describe the center or location of a distribution for a single
quantitative variable are the
The mean for a single quantitative variable is the numerical average of the data values:
To express the calculation of the mean in a mathematical formula, we let
number of data cases in a dataset and
represent the numerical values for the
quantitative variable of interest.
The Greek letter Σ is used as a shorthand for adding all the x values.
As with a proportion, we use different notation to indicate whether a mean summarizes the
data from a sample or a population.
Notation for a Mean
The mean of a sample is denoted and read “x-bar.”
The mean of a population is denoted μ, which is the Greek letter “mu.”
For a random sample of 50 seniors from a large high school, the average SAT (Scholastic
Aptitude Test) score was 582 on the Math portion of the test.
Nearly 1.6 million students in the class of 2010 took the SAT,22 and the average score
overall on the Math portion was 516
The mean of 582 represents the mean of a sample, so we use the notation
mean, and we have
The mean of 516 represents the mean for everyone who took the exam in the class of 2010,
so we use the notation
for the population mean, and we have
The median is another statistic used to summarize the center of a set of numbers. If the
numbers in a dataset are arranged in order from smallest to largest, the median is the middle
value in the list. If there are an even number of values in the dataset, then there is not a unique
middle value and we use the average of the two middle values.
The median of a set of data values for a single quantitative variable, denoted m, is
the middle entry if an ordered list of the data values contains an odd number of entries, or
the average of the middle two values if an ordered list contains an even number of entries.