http://www.ck12.org Chapter 6. Normal Distribution Curves
number of hours that students use a cell phone in 1 day, and we relied upon the sum of the variations to give us some
pertinent information, the only thing that we would learn is that all the students who participated in the survey use a
cell phone for the exact same number of hours each day. We know that this is not true, because the survey does not
show all the responses as being the same. In order to ensure that these variations do not lose their significance when
added, the variation values are squared prior to calculating their sum.
What we need for a normal distribution is a measure of spread that is proportional to the scatter of the data,
independent of the number of values in the data set and independent of the mean. The spread will be small when
the data values are consistent, but large when the data values are inconsistent. The reason that the measure of
spread should be independent of the mean is because we are not interested in this measure of central tendency, but
rather, only in the spread of the data. For a normal distribution, the standard deviation fits the above profile for an
appropriate measure of spread, and this value can be calculated for the set of data.
The standard deviation of a data set is always a positive value.
Example A
A company wants to test its exterior house paint to determine how long it will retain its original color before fading.
The company mixes 2 brands of paint by adding different chemicals to each brand. 6 one-gallon cans are made for
each paint brand, and the results are recorded for every gallon of each brand of paint. The following are the results
obtained in the laboratory:
TABLE6.14:
Brand A (Time in months) Brand B (Time in months)
15 40
65 50
55 35
35 40
45 45
25 30Calculate the standard deviation for each brand of paint. Which brand has more variable data? These are both small
populations.
TABLE6.15: Brand A
x (x−μ) (x−μ)^2
15 − 25 625
65 25 625
55 15 225
35 − 5 25
45 5 25
25 − 15 225