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(Day 1)
So far, we have used histograms to represent
the overall shape of a distribution. Now
smooth curves can be used:
If the curve is symmetric, single peaked, and
bell-shaped, it is called a normal curve.
Plot the data: usually a histogram or a stem
Look for overall pattern
Choose either 5 number summary or “Mean
and Standard Deviation” to describe center
and spread of numbers
◦ 5 number summary used when there are outliers
and graph is skewed; center is the median.
◦ Mean and Standard Deviation used when there are
no outliers and graph is symmetric; center is the
Now, if the overall pattern of a large number of
observations is so regular, it can be described by a
normal curve.
The tails of normal curves fall off quickly.
There are no outlier
There are no outliers.
The mean and median are the same number,
located at the center (peak) of graph.
Most histograms show the “counts” of
observations in each class by the heights of
their bars and therefore by the area of the
◦ (12 = Type A)
Curves show the “proportion” of observations
in each region by the area under the curve.
The scale of the area under the curve equals
1. This is called a density curve.
◦ (0.45 = Type A)
Median: “Equal-areas” point – half area is to the
right, half area is to the left.
Mean: The balance point at which the curve
would balance if made of a solid material (see
next slide).
Area: ¼ of area under curve is to the left of
Quartile 1, ¾ of area under curve is to the left of
Quartile 3. (Density curves use areas “to the
Symmetric: Confirms that mean and median are
Skewed: See next slide.
The mean of a skewed distribution is pulled
along the long tail (away from the median).
Uniform Distributions (height = 1)
If the curve is a normal curve, the standard
deviation can be seen by sight. It is the point
at which the slope changes on the curve.
A small standard
deviation shows
a graph which is
less spread out,
more sharply
Carl Gauss used standard deviations to
describe small errors by astronomers and
surveyors in repeated careful measurements.
A normal curve showing the standard
deviations was once referred to as an “error
The 68-95-99.7 Rule shows the area under
the curve which shows 1, 2, and 3 standard
deviations to the right and the left of the
center of the curve…more accurate than by
More about 68-95-99.7 Rule, z-scores, and
We will be doing group activities. Please
bring your calculators and books!!!
Homework: None… 