Statistics Background
Statistics is a mathematical field that allows us to summarize and analyze data. The two major
subdivisions are descriptive statistics and inferential statistics. In descriptive st at istics, the main go al is
data reduction. That is, large amounts of data are summarized, typically with graphs or tables and by
calculating measures of central tendency and variability. Although these summaries are convenient for
simplifying data and making patterns more obvious, it should be recognized that much of the detail is lost
during summarizing. Descriptive statistics are employed when the whole population can be sampled.
When t he whole population cannot be sampled (the typical situation), a sample (or several) is
withdrawn from the population and inferences about the population are made from the characteristics of
the sample. Thus, it is assumed that a sample is a go od represent ative of the population. However, this
is not always the case and statistics also allows us to decrease our uncertainty about drawing conclusions
from incomplete data.
Measures of Central Tendency
A distribution is created when data variables are arranged in order from highest to lowest,
typically represented graphically as a frequency distribution, plotting the value of the variable versus the
frequency of that value’s occurrence. It was shown quite a long time ago that if the number of data points
is very large, the distribution usually becomes a normal or Gaussian distribution (the bell-shaped curve).
Researchers can learn a lot from these plots, including the shape of the distribution, the range of values,
and t he most common value. Measures of central tendency, t hat is the values that the distribut ion seems
to cluster around, are some of the most commonly calculated statistics.
The mean is one of the most used statistics in all manners of research. It is the arithmetic average
of values, calculated as the sum of all values divided by the number of values. Although the mean
provides a simple summary of a distribution, it doesn’t indicate anything about the range of values. The
dist ribution may have a very tight dist ribu tio n around the mean or may be so spread o ut that a peak is
hard to identify. Another sho rt co ming of the mean is that it is sensitive to extreme values. A single
outlying data point can skew the distribution so that the mean no longer represents the peak of the curve.
Because of this, other descriptors of the central tendency may actually be more useful.
The mode is the most frequently occurring value. It’s where the distribution displays a peak (or
peaks, in the case of a multi-modal distr ibut ion). It is the only descr iptor of central t endency possible
with nominal dat a. (Nominal data classifies items into mutually exclusive groups and can only be
classified as equal or not equal, for example, “male” or “female.”)
The median is the midpoint of the distribution, with half of the values on either side of it.
Another way to say this is that the median represents the 50 percentile. The median can tell us about the
th
shape of the distribution. In a normal distribution, the mean and median (and mode) are the same. If the
distribution is skewed, the mean is closer to the mode than is the median. The median is usually the best
measure of central tendency in a skewed distribution, for example the salaries in the NBA where a few
people earn much, much more than the others.
The Central Limit Theorem
The Central Limit Theorem states that t he means o f independent ly drawn r andom samples will
be approximately normally distributed, if the sample size is large eno ugh. In o ther words, even if a
