Gifted/ Accelerated Math 3
Name:_____________________________
Notes: Population Parameters/ Standard Deviations/ Empirical Rule
Date:____________
Population Parameters


A parameter is a numerical summary of the ______________.
o A parameter is a fixed number, but in practice we do not know its value because we cannot
examine the entire population.
o The total group of individuals about whom you want to make conclusions.
A statistic is a numerical summary of a ______________ taken from the population.
o The value of the statistics is known when we have taken a sample, but it can change from
sample to sample.
o We often use a statistic to estimate an unknown parameter.
o A subset of the population for whom you actually have data.
Remember:
Proportion
Mean
Standard deviation
Number
Parameter
P

“mu”
 “sigma”
N
p̂
x
Statistic
“p-hat”
“x-bar”
s
N
Standard Deviation

Most popular measure of spread….what is spread?





It describes a typical value of how far the data fall from the mean.
A deviation of an observation x from the mean, the difference between the observation and the mean.
The sum of the deviations always equals zero.
The greater the spread of the data, the larger the standard deviation.
The standard deviation equals zero only when all the observations take the same value.
The standard deviation of n observations
of a sample is:
of a population is:
__
s
 ( x  x )2
n 1
sum of squared deviations

sample size - 1

 (x  )
n
2

sum of squared deviations
sample size
Example 1: A financial analyst’s sample of six companies’ book value were
$25, $7, $22, $33, $18, $15.
The sample mean and sample standard deviation are (approximately):
Example 2: A sample of underweight babies was fed a special diet and the following weight gains (lbs)
were observed at the end of three month.
6.7, 2.7, 2.5, 3.6, 3.4, 4.1, 4.8, 5.9, 8.3
The mean and standard deviation are:
Empirical Rule

Using the mean and standard
deviation, we can form
intervals that contain certain
percentages (approximately) of
the data.
Empirical Rule: If a distribution of data is bell-shaped then approximately:
__

68% of the observations fall within 1 standard deviation of the mean, that is ( x  s) or (    ) .

95% of the observations fall within 2 standard deviations of the mean, that is ( x  2s) or (   2 ) .

99.7% or nearly all observations fall within 3 standard deviations of the mean, that is ( x  3s) or
(   3 ) .
__
__
Example 3: A web site for prospective students at a university states that fulltime students spend an average
of $250 for textbooks each semester. The standard deviation of the amounts fulltime students spend for
textbooks per semester is known to be $50.
A. If the $250 value given by the university is correct, do many students spend less than $150 on textbooks
each semester? Explain.
B. If the $250 value given by the university is correct, what are the largest and smallest amounts virtually all
students spend on textbooks per semester? Explain.
C. While buying your books at the bookstore, you notice that the student ahead of you spends $480 for
textbooks that semester. Does this cause you to suspect that the $250 value reported by the university is
incorrect? Explain.
Gifted/ Accelerated Math 3
Name:_____________________________
HW: Population Parameters/ Standard Deviations/ Empirical Rule
Date:____________
1. There are these 8 numbers in a set 9.4, 9.9, 9.9, 9.9, 10.0, 10.2, 10.2, and 10.5. The mean and standard
deviation of the 8 numbers are 10.0 and 0.3 respectively, what percent of the 8 numbers are within 1
standard deviation of the mean?
a. 90%
b. 85%
c. 80%
d. 75%
e. 70%
2. Consider the following three population data sets A, B and C.
A = {9,10,11,7,13}
B = {10,10,10,10,10}
C = {1,1,10,19,19}
a) Calculate the mean of each data set.
b) Calculate the standard deviation of each data set.
c) Which set has the largest standard deviation?
d) Is it possible to answer question c) without calculations of the standard deviation?
3. If the standard deviation of a given data set is equal to zero, what can we say about the data values
included in the given data set?
4. The frequency table of the monthly salaries of 20 people is shown below.
salary(in $)
3500
4000
4200
4300
frequency
5
8
5
2
a) Calculate the mean of the salaries of the 20 people.
b) Calculate the standard deviation of the salaries of the 20 people.
5. The age distribution of a sample of 5000 persons is bell-shaped with a mean of 40 years and a standard
deviation of 12 years. Determine the approximate percentage of people who are 16 to 64 years old.
6. The prices of all college textbooks follows a bell-shaped distribution with a mean of $105 and a
standard deviation of $20.
a. Using the empirical rule, find the percentage of all college textbooks with their prices between
i. $85 & $125
ii. $65 & $145
b. Using the empirical rule, find the interval that contains the prices of 99.7% of college
textbooks.
7. A researcher is studying the hatching Yellow Bellied Sapsuckers. She has a study group of 40 eggs. A
similar study has shown that the eggs’ hatching times should have a mean of 42 days and a variance of
9 days (note that variance is (std dev)2). Her thesis advisor has suggested that the distribution of
hatching times will have the same shape as that of the Purple Throated Tree-tappers, which is “mound
shaped”. If the researcher is willing to assume that her advisor is correct, how may eggs will have
hatching times between 36 and 48 days?
(a) Approximately 55%, or around 22 eggs.
(b) Approximately 68%, or around 27 eggs.
(c) Approximately 75%, or around 30 eggs.
(d) Approximately 95%, or around 38 eggs.
(e) Almost all of the eggs.