Download 18F. Variance and Standard Deviation The variance and closely

18F. Variance and Standard Deviation
Math B30(IB)
The variance and closely-related standard deviation are measures of how
spread out a distribution is. They are more helpful than the range and IQR
because they are calculated by using all data, not just two values.
The variance is computed as the average squared deviation (difference) of
each number from its mean. For example, for the numbers 1, 2, and 3, the mean
is 2 and the variance is
The formula for the variance
in a sample is:
Due to the fact that a sample size is relatively small, the variance of a sample
is considered to be a biased estimate. That is, it tends to overestimate or
underestimate the true value.
To find an unbiased estimate of the variance, we must use the variance of a
. To accomplish this, we use to estimate the mean of the
population
, is incorporated
population, now noted as μ. The variance of the sample,
into the 'new' formula as well. The formula for the variance in a population is:
product of and the sample variance
So what is standard deviation? Well, it's the square root of the variance! It is
the most commonly used measure of spread.
Sample Standard Deviation:
also noted as being the "unbiased estimate of ". Population Standard Deviation:
CAREFUL! Only is in the ! It must be noted that the standard deviation is less useful when a data set
(distribution) includes an outlier. This outlier has an affect on the mean, which
in turn affects the standard deviation, causing it to be less representative of the
distribution.
Example of Sample
.
HMWK: p. 486 #1, 4
Example of Population
HMWK: p. 488 #7, 8
Standard Deviation For Grouped Data
HMWK: p. 489 #11, 14
18G. The Significance of Standard Deviation
The Normal Curve
Normal distributions are bell shaped and completely symmetric about a vertical line
•
passing through the mean of the data.
The mean, median and mode of a set of normally distributed data are the same.
•
68% of the data lies within 1 standard deviation on either side of the mean, that is,
•
34% per side.
95% of the data lies within 2 standard deviations either side of the mean, that is,
•
47.5% per side.
0.3% of the data lies more than 3 standard deviations away on either side of the
•
mean, that is, 0.15% per side.
Note: The normal curve is NOT closed on each end. The curve only approaches
the x-axis.
Example: The speed of 1000 cars was recorded by photo radar. If the data collected
was normally distributed with a mean of 105 km/h and a standard deviation of 10
km/h, draw an appropriate normal curve and determine:
a) the percentage of cars going between 105 and 115 km/h.
b) the percentage of cars going less than 95 km/h.
c) the NUMBER of cars going between 115 km/h and 125 km/h.
HMWK: p. 491 #1, 3, 4
.