Using the 68-95-99.7 Rule
Normal Quantile Plots
Learning Objectives
By the end of this lecture, you should be able to:
– Do various calculations involving areas under the density curve using
the 68-95-99.7 rule
– Identify the mathematical technique used to help confirm (thought not
guarantee!) that our distribution is indeed Normal.
A few numbers worth memorizing
(though not just yet)
•
Because we use the Normal distribution SO much, it is worth memorizing the
approximate areas from the Normal table that correspond to a few different z-scores.
– I say approximate, because the values are rounded off.
•
Look at the areas shown here – but don’t memorize them just yet.
z = -2
z = -1
z = +1
z = +2




about 2.2%
about 16%
about 84%
about 98%
• What I do want you to memorize are the 3 numbers shown in a famous ‘rule’
on the next slide.
The 68-95-99.7% Rule for Normal Distributions

This is essentially a “shortcut” for a mental ballpark
of the areas under the normal curve. It is definitely
worth memorizing.

The area between -1 and +1 standard deviations
corresponds to about 68% of the observations.
The
area between -2 and +2 standard deviations
corresponds to about 95% of the observations.

The area between -3 and +3 standard deviations
corresponds to about 99.7% of the observations.
You WILL be asked to use these numbers on quizzes and exams.
Please note that on your exams you will not be provided with the three numbers (68, 95, 99.7).
Examples: The 68-95-99.7% ‘Shortcut’ Rule for Normal Distributions
Now let’s play around with these numbers by
answering some questions. All numbers refer to z-scores
(i.e. standard deviations):
•
What percentage of observations lie between -1 and +1?
•
Answer: As we just discussed, the number of observations
between -1 and +1 standard deviations is 68%.
•
What percentage lie between 0 and +1?
•
Answer: Recall that z=0 represents 50%. So, if -1 to +1 is
68%, then 0 to +1 is half of that, which is 34%.
•
This is an important one. Make sure you
understand how to do it!!
•
There are a few ways to think of it: Look at the
area between z=0 (the black line) and z=+1. Note
that is is half of the area between -1 and +1.
•
If you need to visualize it (and you should!!), then
shade in the area between z=0 and z=+1.
•
What percentage of observations lie below +1?
•
Answer: To do this, look at your z=0 line. Make sure you
recognize that the area to the left of z=0 represents 50% of
observations. Now, how many observations are between 0
and +1? Recall from the previous question that this is
34%.Therefore, from 0 to +1 = 34, and below 0 is 50, so
the area to the left of +1 represents 84% of observations.
The z=0 line (black line) is very
helpful in doing many of these
calculations.
Examples: The 68-95-99.7% ‘Shortcut’ Rule for Normal Distributions
More examples:
•
What percentage of observations lies between -2 and +1?
•
Answer: Use your midline! I would solve this by adding the area
between -2 and 0 (half of 95%) to the area between 0 and +1 (half
of 68%)  47.5%+ 34% = 81.5%
•
What percentage of observations lies between 0 and +3?
•
Answer: Half of the area between -3 and +3 (99.7) which is 49.85%.
•
What percentage of observations lies below -2?
•
Answer: While this too can be answered in a few different ways, I
would like you to make sure you can do it this way:
•
Look at the area between -2 and +2. Our ‘shortcut’ tells us
that this contains 95% of observations.
•
This means that the area above +2 and below -2 together
compromise 5% of observations. So the area above +2 =
2.5% of observations, and the area below -2 also comprises
2.5% of observations.
•
Answer: 2.5%
•
What percentage of observations lies above +3?
•
Answer: Use the same technique as was just discussed:
•
Between -3 and +3 makes up 99.7.
•
Therefore below -3 and above +3 makes up 0.3%.
•
Therefore below -3 is 0.15% and above +3 = 0.15%
Examples: The 68-95-99.7% ‘Shortcut’ Rule for Normal Distributions
One more!
•
What percentage of observations lies below +2 standard deviations?
•
Answer: Repeat the process from before to determine the area on
either side of +2 and -2. That value was 2.5%. If 2.5% of values lie above
+2, then 97.5% of observations lie below it.
•
Answer: 97.5%
The 68-95-99.7% ‘Shortcut’ Rule for Normal Distributions
•
What percentage of women are
between 62 and 67 inches tall?
•
Answer: Corresponds to -1 to +1 SDs,
Inflection point
that is, about 68%
•
What is the range of heights between
which about 95% of women fall?
•
Answer: About -2 to +2 SDs, so, about
59.5 to 69.5 inches tall.
•
What is the range of heights between
which nearly all (over 99%) of women
fall?
•
Answer: A quick answer would simply to
pick the -3 to +3 SD range (57-72).
mean µ = 64.5
standard deviation s = 2.5
N(µ, s) = N(64.5, 2.5)
The 68-95-99.7% ‘Shortcut’ Rule for Normal Distributions
More Examples:
•
What percentage are taller than
67 inches?
•
Inflection point
Answer: If 68% of all women are
between 62 and 67 inches tall, this
means that 32% are outside of
that range. In other words, 16%
are shorter than 62 inches, and
16% are taller than 67.
•
What percentage are shorter than
59.5 inches?
•
Answer: If 95% of all women are
between 59.5 and 69.5”, then 5%
are outside of that range. In other
words, 2.5% are shorter than 59.5
and 2.5% are taller than 69.5”.
mean µ = 64.5
standard deviation s = 2.5
N(µ, s) = N(64.5, 2.5)
Shortcut Rule or Z-Table?
• Students have often been confused as to which should be
used.
• Whenever possible, use your z-table as you will get a much
more accurate result. In particular, if you are given z-scores that
are not anywhere near whole numbers (e.g. 2.332), then there
is no shortcut to use! The shortcut can only be used with whole
(integer) numbers between -3 and +3.
• The main purpose of learning the ‘shortcut’ rule (in addition to
the fact that they come up on all kinds of exams), is to
encourage you develop an undersatnding of what you are
trying to do rather than just jumping to calculators and ztables.
• For this course, you will be asked to do both.
Is the distribution truly Normal?
•
Deciding whether data does indeed show a Normal (or, close to Normal) distribution is a
very important question.
•
All the examples we’ve been discussing above involving z-scores assume that the data is
Normal. If the data was not Normal, all of our answers and calculations would be flawed.
•
Recall that there are many other types of distributions that are not Normal. Some examples include
skewed, bimodal, Binomial (later in the quarter), Poisson, etc, etc
•
Each type of distribution has its own characteristic formulas, calculations, inference
techniques, etc. Again, because the Normal distirbution is one of the most commonly
encountered distributions, we will spend lots of time discussing it.
•
So how to you decide if a distribution is Normal?
• You might be tempted to say “look at a graph”. And this is not entirely false:
When examining data, a chart is a great (if not the BEST) place to start!
• However, as humans, we are easily fooled. There are many histograms (and
related density curves) that look Normal, but in fact, are not.
• Fortunately, we do have a statistical test that can help confirm (thought not
guarantee) that our dataset does indeed appear to be Normal.
Normal Quantile Plot
The Normal Quantile plot is a graph that helps us determine if a distribution is indeed
Normal
It is a mathematical plot that we can create using our statistical software package of
choice.
Here is the method (which is provided for interest only):
1. The data points are ranked and the percentile ranks are converted to z-scores
with Table A. The z-scores are then used for the x axis against which the data
are plotted on the y axis of the normal quantile plot.
2. If the distribution is indeed normal the plot will show a fairly straight line,
indicating a good match between the data and a normal distribution.
3. Systematic deviations from a straight line indicate a non-normal distribution.
Outliers appear as points that are far away from the overall pattern of the plot.
Normal Quantile Plot shows a
good fit to a straight line: the
distribution of rainwater pH
values is close to normal.
Normal quantile plot is not a
straight line. This tells us that the
data do not follow a Normal
distribution.
Normal quantile plots are complex to do by hand, but they are standard
features in most statistical software.
The normal quantile test supports
normality, but does NOT guarantee it!
• Two key points here:
– If the plot is NOT straight, then your data is NOT normal!
– If the plot IS straight, then you have supported the idea
that your dataset is normal. However, you have NOT
guaranteed it!
• This concept (confirming / supportive tests) will come up with various
other statistical concepts down the road. Whenever you encounter them,
you should be sure to make use of them.