Page 70 - Statistics for Dummies

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Part I: Vital Statistics about Statistics
Standard score
The standard score is a slick way to put results in perspective without having
to provide a lot of details — something that the media loves. The standard
score represents the number of standard deviations above or below the mean
(without caring what that standard deviation or mean actually are).
For example, suppose Bob took his statewide 10th-grade test recently and
scored 400. What does that mean? Not much, because you can’t put 400 into
perspective. But knowing that Bob’s standard score on the test is +2 tells you
everything. It tells you that Bob’s score is two standard deviations above
the mean. (Bravo, Bob!) Now suppose Emily’s standard score is –2. In this
case, this is not good (for Emily), because it means her score is two standard
deviations below the mean.
The process of taking a number and converting it to a standard score is
called standardizing. For the details on calculating and interpreting standard
scores when you have a normal (bell-shaped) distribution, see Chapter 9.
Distribution and normal distribution
The distribution of a data set (or a population) is a listing or function showing
all the possible values (or intervals) of the data and how often they occur.
When a distribution of categorical data is organized, you see the number or
percentage of individuals in each group. When a distribution of numerical
data is organized, they’re often ordered from smallest to largest, broken into
reasonably sized groups (if appropriate), and then put into graphs and charts
to examine the shape, center, and amount of variability in the data.
The world of statistics includes dozens of different distributions for categori-
cal and numerical data; the most common ones have their own names. One
of the most well-known distributions is called the normal distribution, also
known as the bell-shaped curve. The normal distribution is based on numeri-
cal data that is continuous; its possible values lie on the entire real number
line. Its overall shape, when the data are organized in graph form, is a sym-
metric bell-shape. In other words, most (around 68%) of the data are cen-
tered around the mean (giving you the middle part of the bell), and as you
move farther out on either side of the mean, you find fewer and fewer values
(representing the downward sloping sides on either side of the bell).
The mean (and hence the median) is directly in the center of the normal dis-
tribution due to symmetry, and the standard deviation is measured by the
distance from the mean to the inflection point (where the curvature of the
bell changes from concave up to concave down). Figure 4-1 shows a graph
of a normal distribution with mean 0 and standard deviation 1 (this distribu-
tion has a special name, the standard normal distribution or Z-distribution).The
shape of the curve resembles the outline of a bell.

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