Objectives For variables with relatively normal distributions Students should know the approximate percent of observations in a set of data that will fall between the mean and 1 sd 2 sd ID: 403375
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Slide1
The Normal Distribution
Objectives:
For variables with relatively normal distributions:
Students should know the approximate percent of observations in a set of data that will fall between the mean and ± 1
sd
, 2
sd
, and 3
sd
Students should be able to determine the range of values that will contain approximately 68%, 95%, and 99% of the observations in a set of data.Slide2
The Normal Distribution
(also called the bell-shaped or
Gaussian distribution)Slide3
The normal distribution is completely defined by the mean and standard
deviation of a set of quantitative data:
The mean determines the
location
of the curve on the x axis of a graphThe standard deviation determines the height of the curve on the y axis
There are an infinite number of normal distributions- one for every possible combination of a mean and standard deviation
The Normal DistributionSlide4
Pr(X
) on the y-axis refers to either frequency or probability.
Examples of Normal DistributionsSlide5
Examples of Normal DistributionsSlide6
Many (but not all) continuous variables are approximately normally
distributed. Generally, as sample size increases, the shape of a frequency distribution becomes more normally distributed.
Normal DistributionsSlide7
When data are normally distributed, the
mode
, median, and mean are
identical
and are located at the center of the distribution.
Frequencyofoccurrence
Normal DistributionsSlide8
Quantitative variables may also have a
skewed distribution
:When distributions are skewed, they have more extreme values in one direction
than the other, resulting in a
long tail on one side of the distribution. The direction of the tail determines whether a distribution is positively or negatively skewed. A positively skewed distribution has a long tail on the right, or positive side of the curve.A negatively skewed
distribution has the tail on the left, or negative side of the curve.
SkewnessSlide9
Normal distribution
Positively skewed distribution
Negatively skewed distribution
Skewed DistributionsSlide10
Range of Observations
For a normally distributed variable:
~68.3% of the observations lie between the mean and
1 standard deviation~95.4% lie between the mean and 2 standard deviations~99.7% lie between the mean and 3 standard deviationsSlide11
Heart Rate Example
For the heart rate data for 84 adults:
Mean HR = 74.0
bpm
SD = 7.5
bpm
Mean
1SD = 74.0
7.5
= 66.5-81.5
bpm
Mean
2SD = 74.0
15.0
= 59.0-89.0
bpm
Mean
3SD = 74.0
22.5
= 51.5-96.5
bpmSlide12
HR Data:
57
/84 (67.9%) subjects are between mean ± 1SD 82/84 (97.6%) are between mean ± 2SD
84/84 (100%) are between mean ± 3SD
Mean
+3 SD+2 SD
+ 1SD
-1 SD
-2 SD
-3 SD
Heart Rate ExampleSlide13
Reference (“Normal”) Ranges in Medicine
The “normal” range in medical measurements is the central 95% of the values for a reference population, and is usually determined from large samples representative of the population.
The central 95% is approximately the mean
2 sd*Some examples of established reference ranges are:
Serum
“
Normal” range
fasting
glucose
70
-110 mg/
dL
sodium
135
-146
mEq
/L
triglycerides
35
-160 mg/
dL
Note
: The value is actually 1.96
sd
but for convenience this is
usually
rounded
to 2 sd.Slide14
The Standard Normal Distribution
A normal distribution with a mean of 0, and
sd
of
1 The distribution is also called the z distributionAny normal distribution can be converted to the standard normal distribution using the z transformation. Each value in a distribution is converted to the number of standard deviations the value is from the mean.
The transformed value is called a z score.Slide15
Once the data are transformed to
z
-scores
, the standard normal distribution can be used to determine areas under the curve for any normal distribution.
Formula for the z transformationSlide16
Example of a z
-transformation
If the population mean heart rate is 74
bpm
, and the standard deviation is 7.5, the z score for an individual with HR = 80 bpm is:
The individual’s HR of 80 bpm
is
0.8 standard deviations above the mean.Slide17
Rule of Thumb #1
The
z-value
can be looked up in a table for the standard normal distribution to determine the lower and upper areas defined by a
z-score of 0.8 (the areas are the lower 78.8% and upper 21.2%)You will not need to calculate z-scores or find corresponding areas under the curve for z
-scores in this class, but you will be expected to know the following:
The important
z
-
scores
to know are
±1.645, ±1.960*, ±2.575
Note
: when calculating by hand, it is OK to round 1.960 to 2 Slide18
Rule of Thumb #2
The total area under the normal distribution curve is 1:
90% of the area is between ± 1.645 sd
95% of the area is between ± 1.960
sd99% of the area is between ± 2.575 sdSlide19
The Normal Distribution & Confidence Intervals
90% of the area is between ± 1.645
sd95% of the area is between ± 1.960
sd
99% of the area is between ± 2.575 sdThese are the most commonly used areas for definingConfidence Intervalswhich are used in inferential statistics to estimate population values from sample data