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Review of Basic Statistics
Parameters and Statistics
• Parameters are characteristics of populations, and are knowable only by taking a census.
• Statistics are estimates of parameters made from samples.
Descriptive Statistics Review
Measures of LocationThe MeanThe MedianThe Mode
Measures of Dispersion
The variance
The standard deviation
MeanThe mean (or average) is the basic measure of location or “central
tendency” of the data.
•The sample mean is a sample statistic.
•The population mean is a population statistic.
x
Sample Mean
n
xx i
Where the numerator is the sum of values of n observations, or:
ni xxxx ...21
The Greek letter Σ is the summation sign
Example: College Class SizeWe have the following sample of data for 5 college classes:
46 54 42 46 32
We use the notation x1, x2, x3, x4, and x5 to represent the number of students in each of the 5 classes:
X1 = 46 x2 = 54 x3 = 42 x4 = 46 x5 = 32
Thus we have:
445
3246425446
554321
xxxxx
n
xx i
The average class size is 44 students
Population Mean ()
N
xi
The number of observations in the population is denoted by the upper case N.
The sample mean is a point estimator of
the population mean
x
Median The median is the value in the middle when the data are arranged in ascending order (from smallest value to largest value).
a. For an odd number of observations the median is the middle value.
b. For an even number of observations the median is the average of the two middle values.
The College Class Size example
First, arrange the data in ascending order:
32 42 46 46 54
Notice than n = 5, an odd number. Thus the median is given by the middle value.
32 42 46 46 54
The median class size is 46
Median Starting Salary For a Sample of 12 Business School Graduates
A college placement office has obtained the following data for 12 recent graduates:
Graduate Starting Salary GraduateStarting Salary
1 2850 7 2890
2 2950 8 3130
3 3050 9 2940
4 2880 10 3325
5 2755 11 2920
6 2710 12 2880
2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325
Notice that n = 12, an even number. Thus we take an average of the middle 2 observations:
2710 2755 2850 2880 2880 2890 2920 2940 2950 3050 3130 3325
Middle two values
First we arrange the data in
ascending order
29052
29202890Median
Thus
Mode The mode is the value that occurs with greatest frequency
Soft Drink Example
Soft Drink Frequency
Coke Classic 19
Diet Coke 8
Dr. Pepper 5
Pepsi Cola 13
Sprite 5
Total 50
The mode is Coke Classic. A mean
or median is meaningless of qualitative data
Using Excel to Compute the Mean, Median, and Mode
Enter the data into cells A1:B13 for the starting salary example.
•To compute the mean, activate an empty cell and enter the following in the formula bar:
=Average(b2:b13) and click the green checkmark.
•To compute the median, activate an empty cell and enter the following in the formula bar:
= Median(b2:b13) and click the green checkmark.
•To compute the mode, activate an empty cell and enter the following in the formula bar:
=Average(b2:b13) and click the green checkmark.
The Starting Salary Example
Mean 2940
Median 2905
Mode 2880
Variance• The variance is a measure of variability that uses all the
data• The variance is based on the difference between each
observation (xi) and the mean ( ) for the sample and μ for the population).x
The variance is the average of the squared differences between the observations and the mean value
For the population:N
xi2
2 )(
For the sample:1
)( 22
n
xxs i
Standard Deviation
• The Standard Deviation of a data set is the square root of the variance.
• The standard deviation is measured in the same units as the data, making it easy to interpret.
Computing a standard deviation
1
)( 2
n
xxs i
For the population:
For the sample:
N
xi2)(
Measures of AssociationBetween two Variables
•Covariance
•Correlation coefficient
Covariance
• Covariance is a measure of linear association between variables.
• Positive values indicate a positive correlation between variables.
• Negative values indicate a negative correlation between variables.
To compute a covariance for variables x and y
N
uyx yixixy
))((
For populations
1
))((
n
yyxxs iixy
For samples
Mortgage Interest Rates and Monthly Home Sales, 1980-2004
3
5
7
9
11
13
15
17
15 35 55 75 95 115
Monthly Home Sales (thousands)
Mor
tgag
e In
tere
st R
ate
(Per
cent
)3.60x
02.9y
n = 299
II I
III
IV
If the majority of the sample points are
located in quadrants II and IV, you have a negative correlation
between the variables—as we do in this case.
Thus the covariance will have a negative sign.
The (Pearson) Correlation Coefficient
A covariance will tell you if 2 variables are positively or
negatively correlated—but it will not tell you the degree of correlation. Moreover, the
covariance is sensitive to the unit of measurement. The correlation coefficient does not suffer from
these defects
The (Pearson) Correlation Coefficient
yx
xyxy
yx
xyxy ss
sr
For populations
For samples
Note that:
11
and
11
xy
xy
r
Correlation Coefficient = 1
0
100
200
300
400
500
0 20 40 60 80 100
Average Speed (MPH)
Dis
tan
ce T
rave
led
in
5
Ho
urs
(M
iles
)
Correlation Coefficient = -1
012345678
0 2 4 6 8
Time Spent Jogging (Hours)
Tim
e S
pen
t S
wim
min
g
(Ho
urs
)
I have 7 hours per week for exercise
Normal Probability Distribution
The normal distribution is by far the most important
distribution for continuous random variables. It is widely
used for making statistical inferences in both the natural
and social sciences.
HeightsHeightsof peopleof peopleHeightsHeights
of peopleof people
Normal Probability DistributionNormal Probability Distribution
It has been used in a wide variety of It has been used in a wide variety of applications:applications:
ScientificScientific measurementsmeasurements
ScientificScientific measurementsmeasurements
AmountsAmounts
of rainfallof rainfall
AmountsAmounts
of rainfallof rainfall
Normal Probability DistributionNormal Probability Distribution
It has been used in a wide variety of It has been used in a wide variety of applications:applications:
TestTest scoresscoresTestTest
scoresscores
The Normal Distribution
22 2/)(
2
1)(
xexf
Where:
μ is the mean
σ is the standard deviation
= 3.1459
e = 2.71828
The distribution is The distribution is symmetricsymmetric, and is , and is bell-shapedbell-shaped.. The distribution is The distribution is symmetricsymmetric, and is , and is bell-shapedbell-shaped..
Normal Probability DistributionNormal Probability Distribution
CharacteristicsCharacteristics
xx
The entire family of normal probabilityThe entire family of normal probability distributions is defined by itsdistributions is defined by its meanmean and its and its standard deviationstandard deviation . .
The entire family of normal probabilityThe entire family of normal probability distributions is defined by itsdistributions is defined by its meanmean and its and its standard deviationstandard deviation . .
Normal Probability DistributionNormal Probability Distribution
CharacteristicsCharacteristics
Standard Deviation Standard Deviation
MeanMean xx
The The highest pointhighest point on the normal curve is at the on the normal curve is at the meanmean, which is also the , which is also the medianmedian and and modemode.. The The highest pointhighest point on the normal curve is at the on the normal curve is at the meanmean, which is also the , which is also the medianmedian and and modemode..
Normal Probability DistributionNormal Probability Distribution
CharacteristicsCharacteristics
xx
Normal Probability DistributionNormal Probability Distribution
CharacteristicsCharacteristics
-10-10 00 2020
The mean can be any numerical value: negative,The mean can be any numerical value: negative, zero, or positive.zero, or positive. The mean can be any numerical value: negative,The mean can be any numerical value: negative, zero, or positive.zero, or positive.
xx
Normal Probability DistributionNormal Probability Distribution
CharacteristicsCharacteristics
= 15= 15
= 25= 25
The standard deviation determines the width of theThe standard deviation determines the width of thecurve: larger values result in wider, flatter curves.curve: larger values result in wider, flatter curves.The standard deviation determines the width of theThe standard deviation determines the width of thecurve: larger values result in wider, flatter curves.curve: larger values result in wider, flatter curves.
xx
Probabilities for the normal random variable areProbabilities for the normal random variable are given by given by areas under the curveareas under the curve. The total area. The total area under the curve is 1 (.5 to the left of the mean andunder the curve is 1 (.5 to the left of the mean and .5 to the right)..5 to the right).
Probabilities for the normal random variable areProbabilities for the normal random variable are given by given by areas under the curveareas under the curve. The total area. The total area under the curve is 1 (.5 to the left of the mean andunder the curve is 1 (.5 to the left of the mean and .5 to the right)..5 to the right).
Normal Probability DistributionNormal Probability Distribution
CharacteristicsCharacteristics
.5.5 .5.5
xx
The Standard Normal Distribution
0
The Standard Normal Distribution is a normal distribution with the
special properties that is mean is zero and its standard deviation is
one.
1
00zz
The letter The letter z z is used to designate the standardis used to designate the standard normal random variable.normal random variable. The letter The letter z z is used to designate the standardis used to designate the standard normal random variable.normal random variable.
Standard Normal Probability DistributionStandard Normal Probability Distribution
Cumulative ProbabilityCumulative Probability
00 11zz
)1( zP
Probability that z ≤ 1 is the area under the curve to the left of 1.
What is P(z ≤ 1)?
Z .00 .01 .02
●
●
●
.9 .8159 .8186 .8212
1.0 .8413 .8438 .8461
1.1 .8643 .8665 .8686
1.2 .8849 .8869 .8888
●
●
To find out, use the Cumulative Probabilities Table for the Standard Normal Distribution
)1( zP
Area under the curveArea under the curve
00zz
211--11
-2
68.25%
95.45%
•68.25 percent of the total area under the curve is within (±) 1 standard deviation from the mean.
•95.45 percent of the area under the curve is within (±) 2 standard deviations of the mean.
Exercise 1
2.46
a) What is P(z ≤2.46)?
b) What is P(z >2.46)?
Answer:
a) .9931
b) 1-.9931=.0069
z
Exercise 2
-1.29
a) What is P(z ≤-1.29)?
b) What is P(z > -1.29)?
Answer:
a) 1-.9015=.0985
b) .9015
Note that, because of the symmetry, the area to the left of -1.29 is the same as the area to the right of 1.29
1.29
Red-shaded area is equal to green- shaded area
Note that:
)29.1(1)29.1( zPzP
z
Exercise 3
0
What is P(.00 ≤ z ≤1.00)?
1
3413.5000.8413.
)0()1()100(.
zPzPzP
P(.00 ≤ z ≤1.00)=.3413
z