blacktriangle Note: NY NGMS will concentrate only on the "Sample Standard Deviation".

bullet Standard Deviation

Standard deviation helps determine how closely data is clustered around the mean value.

definition

Standard deviation shows how much variation (dispersion, spread, scatter) from the mean exists. It represents a "typical" deviation from the mean.

The standard deviation can be thought of as a "standard" way of knowing what is normal (typical), what is very large, and what is very small in the data set.

It is the average amount that data values deviate from the mean value of the data.

If the standard deviation is low (a small number), the data will be clustered close to the mean.
If the standard deviation is high (a large number), the data is spread over a large range of values.

Standard deviation is a popular measure of variability because it returns the original units of measure of the data set. For example, original data containing lengths measured in feet has a standard deviation also measured in feet.

NOTATIONS: Standard deviation may be abbreviated SD or sd.
Mathematically speaking, it is represented by
• the lower case Greet letter σ (sigma) for population sd.
• and the Latin letter S for sample sd.

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bullet Normal Curve and SD


A normal curve is a symmetric,
bell-shaped curve.

The center of the graph is the mean,
and the height and width of the graph are determined by the standard deviation of the data.

The mean, median, and mode have the
same value in a normal curve.


A normal distribution can have any mean and any positive standard deviation.
The mean, , determines the line of symmetry of the graph, and the standard deviation, SD, determines how much the data are spread out.

• When the
standard deviation is small, the curve will be tall and narrow in spread.
• When the
standard deviation is large, the curve will be short and wide in spread.

Most data falls within one standard deviation of the mean.

Normal Curve Empirical Rule:
Approximately
...

• 68% of the data lie within one standard deviation of the mean.

• 95% of the data lies within two standard deviations of the mean.

• 99.7% of the data lies within three standard deviations of the mean.

IQR for a normal curve is
1.34896 x standard deviation.

normalgraphe

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bullet Thinking About Standard Deviation



If you know that the mean on the last math test was 83 and that the standard deviation was 0, what do you know about the test scores?

The weights of 5 cats, in pounds, are 8, 10, 11, 13, 15, and you are told that the sample standard deviation is 2.7.
a) What is the mean weight?

b) What would be a weight of 2 SD above the mean?

c) What would be a weight of 1 SD below the mean?


Ex 1 - ANS: Everyone got the same score of 83.
Ex 2 - ANS: a) 11.4      b) 11.4 + 2(2.7) = 16.8 lbs.    c) 11.4 - 1(2.7) = 8.7 lbs.


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See Calculating Standard Deviation
for calculator skills, formulas and by hand calculations.


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