Chapter 3 Descriptive Statistics
Sample Standard Deviation
Population Mean
Computational Formulas for Population Variance
and Standard Deviation
Sample Mean
Interquartile Range
Q3 – Q1
Sum of Deviations from the Arithmetic Mean is
Always Zero
Computational Formulas for Sample Variance and
Standard Deviation
Mean Absolute Deviation
z Score
Population Variance
Coefficient of Variation
Population Standard Deviation
Mean of Grouped Data
Empirical Rule*
Distance from the Mean
Values within the Distance
68%
95%
99.7%
where
i = the number of classes
f = class frequency
M = class midpoint
N = total frequencies (total number of data values)
*Based on the assumption that the data are
approximately normally distributed.
Chebyshev’s Theorem
Within k standard deviations of the mean,
at least
proportion of the values.
Assumption: k > 1
Sample Variance
Medium of Grouped Data
, lie
where
l = lower endpoint of the class containing the median
w = width of the class containing the median
f = frequency of the class containing the median
F = cumulative frequency of classes preceding the
class containing the median
N = total frequencies (total number of data values)
Formulas for Population Variance and Standard
Deviation of Grouped Data
Original Formula
Computational Version
where
f = frequency
M = class midpoint
N = ∑ , or total of the frequencies of the population
= grouped mean for the population
Formulas for Sample Variance and Standard
Deviation of Grouped Data
Original Formula
Computational Version
where
f = frequency
M = class midpoint
N = ∑ , or total of the frequencies of the population
= grouped mean for the sample
Coefficient of Skewness
where
= coefficient of skewness
= median
Independent Events X, Y
If X and Y are independent events, the following must be
true:
Chapter 4 Probability
Bayes’ Rule
Classical Method of Assigning Probabilities
where
N = total possible number of outcomes of an
experiment
= the number of outcomes in which the event
occurs out of N outcomes
Range of Possible Probabilities
Probability by Relative Frequency of Occurrence
Mutually Exclusive Events X and Y
Independent Events X and Y
Probability of the Complement of A
The mn Counting Rule
For an operation that can be done m ways and a second
operation that can be done n ways, the two operations
can then occur, in order, in mn ways. This rule can be
extended to cases with three or more operations.
General Law of Addition
where X, Y, are events and
of X and Y.
Special Law of Addition
If X, Y are mutually exclusive,
General Law of Multiplication
Special Law of Multiplication
If X, Y are independent,
Law of Conditional Probability
is the intersection
e = 2.718281…
Chapter 5 Discrete Distributions
Mean or Expected Value of a Discrete Distribution
Hypergeometric Formula
where
E(x) = long-run average
x = an outcome
P(x) = probability of that outcome
Variance of a Discrete Distribution
where
x = an outcome
P(x) = probability of a given outcome
= mean
Standard Deviation of a Discrete Distribution
Assumptions of the Binomial Distribution
- The experiment involves n identical trials.
- Each trial has only two possible outcomes denoted as
success or failure.
- Each trial is independent of the previous trials.
- The terms p and q remain constant throughout the
experiment, where the term p is the probability of
getting a success on any one trial and the term q = 1 –
p is the probability of getting a failure on any one
trial.
Binomial Formula
where
n = the number of trials (or the number being
sampled)
x = the number of successes desired
p = the probability of getting a success in one trial
= 1 – p = the probability of getting a failure in one
trial
Mean and Standard Deviation of a Binomial
Distribution
Poisson Formula
where
x = 0, 1, 2, 3, …
= long-run average
where
N = size of the population
n = sample size
A = number of successes in the population
x = number of successes in the sample; sampling is
done without replacement
Chapter 6 Continuous Distributions
Probability Density Function of a Uniform
Distribution
Mean and Standard Deviation of a Uniform
Distribution
Probabilities in a Uniform Distribution
where
z formula
Exponential Probability Density Function
where
x
and e = 2.271828…
Probabilities of the Right Tail of the Exponential
Distribution
where
= sample proportion
n = sample size
p = population proportion
q=1–p
Chapter 7 Sampling and Sampling
Distributions
Determining the Value of k
where
n = sample size
N = population size
k = size of interval for selection
Central Limit Theorem
If samples of size n are drawn randomly from a
population that has a mean of
and a standard
deviation of , the sample means, , are approximately
normally distributed for sufficiently large samples (n
30*) regardless of the shape of the population
distribution. If the population is normally distributed,
the sample means are normally distributed for any
sample size.
From mathematical expectation, it can be shown that
the mean of the sample means is the population mean:
and the standard deviation of the sample means (called
the standard error of the mean) is the standard
deviation of the population divided by the square root
of the sample size:
z Formula for Sample Means
z Formula for Sample Means of a Finite Population
Sample Proportion
where
x = number of items in a sample that have the
characteristic
n = number of items in the sample
z Formula for Sample Proportions for n
n
where
and
Chapter 8 Statistical Inference: Estimation for
Single Populations
100(1 – a)% Confidence Interval to Estimate
Known (8.1)
Sample Size When Estimating
(8.7)
:
Sample Size When Estimating p (8.8)
where
=
the area under the normal curve outside the
confidence interval area
 = the area in one end (tail) of the distribution
outside the confidence interval
Confidence Interval to Estimate
Correction Factor (8.2)
Using the Finite
Confidence Interval to Estimate : Population
Standard Deviation Unknown and the Population
Normally Distributed (8.3)
Confidence Interval to Estimate p (8.4)
where
= sample proportion
= sample size
p = population proportion
n = sample size
Formula for Single Variance (8.5)
Confidence Interval to Estimate the Population
Variance (8.6)
where
p = population proportion
q= 1 – p
E = error of estimation
n = sample size
Chapter 9 Statistical Inference: Hypothesis
Testing for Single Populations
z Test for a Single Mean (9.1)
Formula to Test Hypotheses about
Population (9.2)
t Test for
with a Finite
(9.3)
z Test of a Population Proportion (9.4)
where
= sample proportion
p = population proportion
q=1–p
Formula for Testing Hypotheses about a Population
Variance (9.5)
Chapter 10 Statistical Inference: About Two
Populations
Confidence Interval to Estimate μ1 − μ2 Assuming
the Population Variances are Unknown and Equal
(10.5)
z Formula for the Difference in Two Sample Means
(Independent Samples and Population Variances
Known) (10.1)
t Formula to Test the Difference in two Dependent
Populations (10.6)
where
= mean of population 1
= mean of population 2
= size of sample 1
= size of sample 2
Confidence Interval to Estimate
−
(10.2)
t Formula to Test the Difference in Means Assuming
and
are Equal (10.3)
where
n = number of pairs
d = sample difference in pairs
D = mean population difference
sd = standard deviation of sample difference
d = mean sample difference
Formulas for d and sd (10.7 and 10.8)
where
t Formula to Test the Difference in Means (10.4)
Confidence Interval Formula to Estimate the
Difference in Related Populations, D (10.9)
z Formula for the Difference in Two Population
Proportions (10.10)
Formula for Determining the Critical Value for the
Lower-Tail F (10.14)
where
= proportion from sample 1
= proportion from sample 2
= size of sample 1
= size of sample 2
= proportion from population 1
= proportion from population 2
=
=
z Formula to Test the Difference in Population
Proportions (10.11)
where
and
Confidence Interval to Estimate p1 - p2 (10.12)
F Test for Two Population Variances (10.13)
(12.3) Alternative formula for slope
(12.4) y intercept of the regression line
Sum of squares of error
Chapter 12 Correlation and Simple Regression
Analysis
(12.1) Pearson product-moment correlation
coefficient
Standard error of the estimate
Equation of the simple regression line
(12.5) Coefficient of determination
Sum of squares
Computational formula for r2
t test of slope
(12.2) Slope of the regression line
(12.6) Confidence interval to estimate E(yx) for a
given value of x
(12.7) Prediction interval to estimate y for a given
value of x