Chapter Summary – Descriptive stats
I. A frequency table is a grouping of qualitative data into mutually exclusive classes showing the number of observations in each class.
II. A relative frequency table shows the fraction of the number of frequencies in each class.
III. A bar chart is a graphic representation of a frequency table.
IV. A pie chart shows the proportion each distinct class represents of the total number of frequencies.
V. A frequency distribution is a grouping of data into mutually exclusive classes showing the number of observations in each class.
A. The steps in constructing a frequency distribution are:
1. Decide on the number of classes.
2. Determine the class interval.
3. Set the individual class limits.
4. Tally the raw data into classes.
5. Count the number of tallies in each class.
B. The class frequency is the number of observations in each class.
C. The class interval is the difference between the limits of two
consecutive classes.
D. The class midpoint is halfway between the limits of consecutive
classes.
VI. A relative frequency distribution shows the percent of observations in
each class.
VII. There are three methods for graphically portraying a frequency
distribution.
A. A histogram portrays the number of frequencies in each class in the
form of a rectangle.
B. A frequency polygon consists of line segments connecting the points
formed by the intersection of the class midpoint and the class
frequency.
C. A cumulative frequency distribution shows the number or percent of
observations below given values.
CHAPTER SUMMARY
I. A measure of location is a value used to describe the center of a set of
data.
A. The arithmetic mean is the most widely reported measure of location.
1. It is calculated by adding the values of the observations and dividing by the
total number of observations.
a. The formula for a population mean of ungrouped or raw data is
[3–1]
b. The formula for the mean of a sample is
[3–2]
c. The formula for the sample mean of data in a frequency distribution is
[3–12]
2. The major characteristics of the arithmetic mean are:
a. At least the interval scale of measurement is required.
b. All the data values are used in the calculation.
c. A set of data has only one mean. That is, it is unique.
d. The sum of the deviations from the mean equals 0.
B. The weighted mean is found by multiplying each observation by its
corresponding weight.
1. The formula for determining the weighted mean is
[3–3]
2. It is a special case of the arithmetic mean.
C. The median is the value in the middle of a set of ordered data.
1. To find the median, sort the observations from smallest to largest and identify the middle value.
2. The major characteristics of the median are:
a. At least the ordinal scale of measurement is required.
b. It is not influenced by extreme values.
c. Fifty percent of the observations are larger than the median.
d. It is unique to a set of data.
D. The mode is the value that occurs most often in a set of data.
1. The mode can be found for nominal-level data.
2. A set of data can have more than one mode.
E. The geometric mean is the nth root of the product of n positive values.
1. The formula for the geometric mean is
[3–4]
2. The geometric mean is also used to find the rate of change from one period to another.
[3–5]
3. The geometric mean is always equal to or less than the arithmetic mean.
II. The dispersion is the variation or spread in a set of data.
A. The range is the difference between the largest and the smallest value in a set of data.
1. The formula for the range is
[3–6]
2. The major characteristics of the range are:
a. Only two values are used in its calculation.
b. It is influenced by extreme values.
c. It is easy to compute and to understand.
B. The mean absolute deviation is the sum of the absolute values of the deviations from the mean divided by the number of observations.
1. The formula for computing the mean absolute deviation is
[3–12]
2. The major characteristics of the mean absolute deviation are:
a. It is not unduly influenced by large or small values.
b. All observations are used in the calculation.
c. The absolute values are somewhat difficult to work with.
C. The variance is the mean of the squared deviations from the arithmetic mean.
1. The formula for the population variance is
[3–8]
2. The formula for the sample variance is
[3–10]
3. The major characteristics of the variance are:
a. All observations are used in the calculation.
b. It is not unduly influenced by extreme observations.
c. The units are somewhat difficult to work with; they are the original units squared.
D. The standard deviation is the square root of the variance.
1. The major characteristics of the standard deviation are:
a. It is in the same units as the original data.
b. It is the square root of the average squared distance from the mean.
c. It cannot be negative.
d. It is the most widely reported measure of dispersion.
2. The formula for the sample standard deviation is
[3–11]
3. The formula for the standard deviation of grouped data is
[3–13]
III. We interpret the standard deviation using two measures.
A. Chebyshev’s theorem states that regardless of the shape of the distribution, at least 1 – 1/k2 of the observations will be within k standard deviations of the mean, where k is greater than 1.
B. The Empirical Rule states that for a bell-shaped distribution about 68 percent of the values will be within one standard deviation of the mean, 95 percent within two, and virtually all within three.
Probability
I. A probability is a value between 0 and 1 inclusive that represents the likelihood a particular event will happen.
A. An experiment is the observation of some activity or the act of taking some measurement.
B. An outcome is a particular result of an experiment.
C. An event is the collection of one or more outcomes of an experiment.
II. There are three definitions of probability.
A. The classical definition applies when there are n equally likely outcomes to an experiment.
B. The empirical definition occurs when the number of times an event happens is divided by the number of observations.
C. A subjective probability is based on whatever information is available.
III. Two events are mutually exclusive if by virtue of one event happening the other cannot happen.
IV. Events are independent if the occurrence of one event does not affect the occurrence of another event.
V. The rules of addition refer to the union of events.
A. The special rule of addition is used when events are mutually exclusive.
B. The general rule of addition is used when the events are not mutually exclusive.
C. The complement rule is used to determine the probability of an event happening by subtracting the probability of the event not happening from 1.
VI. The rules of multiplication refer to the product of events.
A. The special rule of multiplication refers to events that are independent.
B. The general rule of multiplication refers to events that are not i independent.
C. A joint probability is the likelihood that two or more events will happen at the same time.
D. A conditional probability is the likelihood that an event will happen, given that another event has already happened.
E. Bayes’ theorem is a method of revising a probability, given that additional information is obtained. For two mutually exclusive and collectively exhaustive events:
VII. There are three counting rules that are useful in determining the number of outcomes in an experiment.
A. The multiplication rule states that if there are m ways one event can happen and n ways another event can happen, then there are mn ways the two events can happen.
B. A permutation is an arrangement in which the order of the objects selected from a specific pool of objects is important.
C. A combination is an arrangement where the order of the objects selected from a specific pool of objects is not important.
Summary notesExpected values and random variables
I. A random variable is a numerical value determined by the outcome of an experiment.
II. A probability distribution is a listing of all possible outcomes of an experiment and the probability associated with each outcome.
A. A discrete probability distribution can assume only certain values. The main features are:
1. The sum of the probabilities is 1.00.
2. The probability of a particular outcome is between 0.00 and 1.00.
3. The outcomes are mutually exclusive.
B. A continuous distribution can assume an infinite number of values within a specific range.
III. The mean and variance of a probability distribution are computed as follows.
A. The mean is equal to:
B. The variance is equal to:
IV. The binomial distribution has the following characteristics.
A. Each outcome is classified into one of two mutually exclusive categories.
B. The distribution results from a count of the number of successes in a fixed number of trials.
C. The probability of a success remains the same from trial to trial.
D. Each trial is independent.
E. A binomial probability is determined as follows:
F. The mean is computed as:
G. The variance is
V. The hypergeometric distribution has the following characteristics.
A. There are only two possible outcomes.
B. The probability of a success is not the same on each trial.
C. The distribution results from a count of the number of successes in a fixed number of trials.
D. It is used when sampling without replacement from a finite population.
E. A hypergeometric probability is computed from the following equation:
VI. The Poisson distribution has the following characteristics.
A. It describes the number of times some event occurs during a specified interval.
B. The probability of a “success” is proportional to the length of the interval.
C. Nonoverlapping intervals are independent.
D. It is a limiting form of the binomial distribution when n is large and π is small.
E. A Poisson probability is determined from the following equation:
F. The mean and the variance are:
Summary notes probability distributions
I. The uniform distribution is a continuous probability distribution with the following characteristics.
A. It is rectangular in shape.
B. The mean and the median are equal.
C. It is completely described by its minimum value a and its maximum value b.
D. It is also described by the following equation for the region from a to b:
E. The mean and standard deviation of a uniform distribution are computed as follows:
II. The normal probability distribution is a continuous distribution with the following characteristics.
A. It is bell-shaped and has a single peak at the center of the distribution.
B. The distribution is symmetric.
C. It is asymptotic, meaning the curve approaches but never touches the X-axis.
D. It is completely described by its mean and standard deviation.
E. There is a family of normal probability distributions.
1. Another normal probability distribution is created when either the mean or the standard deviation changes.
2. The normal probability distribution is described by the following formula:
III. The standard normal probability distribution is a particular normal distribution.
A. It has a mean of 0 and a standard deviation of 1.
B. Any normal probability distribution can be converted to the standard normal probability distribution by the following formula.
C. By standardizing a normal probability distribution, we can report the distance of a value from the mean in units of the standard deviation.
IV. The normal probability distribution can approximate a binomial distribution under certain conditions.
A. π and n(1 – π) must both be at least 5.
1. n is the number of observations.
2. π is the probability of a success.
B. The four conditions for a binomial probability distribution are:
1. There are only two possible outcomes.
2. π remains the same from trial to trial.
3. The trials are independent.
4. The distribution results from a count of the number of successes in a fixed number of trials.
C. The mean and variance of a binomial distribution are computed as follows:
D. The continuity correction factor of .5 is used to extend the continuous value of X one-half unit in either direction. This correction compensates for approximating a discrete distribution by a continuous distribution.
I. A frequency table is a grouping of qualitative data into mutually exclusive classes showing the number of observations in each class.
II. A relative frequency table shows the fraction of the number of frequencies in each class.
III. A bar chart is a graphic representation of a frequency table.
IV. A pie chart shows the proportion each distinct class represents of the total number of frequencies.
V. A frequency distribution is a grouping of data into mutually exclusive classes showing the number of observations in each class.
A. The steps in constructing a frequency distribution are:
1. Decide on the number of classes.
2. Determine the class interval.
3. Set the individual class limits.
4. Tally the raw data into classes.
5. Count the number of tallies in each class.
B. The class frequency is the number of observations in each class.
C. The class interval is the difference between the limits of two
consecutive classes.
D. The class midpoint is halfway between the limits of consecutive
classes.
VI. A relative frequency distribution shows the percent of observations in
each class.
VII. There are three methods for graphically portraying a frequency
distribution.
A. A histogram portrays the number of frequencies in each class in the
form of a rectangle.
B. A frequency polygon consists of line segments connecting the points
formed by the intersection of the class midpoint and the class
frequency.
C. A cumulative frequency distribution shows the number or percent of
observations below given values.
CHAPTER SUMMARY
I. A measure of location is a value used to describe the center of a set of
data.
A. The arithmetic mean is the most widely reported measure of location.
1. It is calculated by adding the values of the observations and dividing by the
total number of observations.
a. The formula for a population mean of ungrouped or raw data is
[3–1]
b. The formula for the mean of a sample is
[3–2]
c. The formula for the sample mean of data in a frequency distribution is
[3–12]
2. The major characteristics of the arithmetic mean are:
a. At least the interval scale of measurement is required.
b. All the data values are used in the calculation.
c. A set of data has only one mean. That is, it is unique.
d. The sum of the deviations from the mean equals 0.
B. The weighted mean is found by multiplying each observation by its
corresponding weight.
1. The formula for determining the weighted mean is
[3–3]
2. It is a special case of the arithmetic mean.
C. The median is the value in the middle of a set of ordered data.
1. To find the median, sort the observations from smallest to largest and identify the middle value.
2. The major characteristics of the median are:
a. At least the ordinal scale of measurement is required.
b. It is not influenced by extreme values.
c. Fifty percent of the observations are larger than the median.
d. It is unique to a set of data.
D. The mode is the value that occurs most often in a set of data.
1. The mode can be found for nominal-level data.
2. A set of data can have more than one mode.
E. The geometric mean is the nth root of the product of n positive values.
1. The formula for the geometric mean is
[3–4]
2. The geometric mean is also used to find the rate of change from one period to another.
[3–5]
3. The geometric mean is always equal to or less than the arithmetic mean.
II. The dispersion is the variation or spread in a set of data.
A. The range is the difference between the largest and the smallest value in a set of data.
1. The formula for the range is
[3–6]
2. The major characteristics of the range are:
a. Only two values are used in its calculation.
b. It is influenced by extreme values.
c. It is easy to compute and to understand.
B. The mean absolute deviation is the sum of the absolute values of the deviations from the mean divided by the number of observations.
1. The formula for computing the mean absolute deviation is
[3–12]
2. The major characteristics of the mean absolute deviation are:
a. It is not unduly influenced by large or small values.
b. All observations are used in the calculation.
c. The absolute values are somewhat difficult to work with.
C. The variance is the mean of the squared deviations from the arithmetic mean.
1. The formula for the population variance is
[3–8]
2. The formula for the sample variance is
[3–10]
3. The major characteristics of the variance are:
a. All observations are used in the calculation.
b. It is not unduly influenced by extreme observations.
c. The units are somewhat difficult to work with; they are the original units squared.
D. The standard deviation is the square root of the variance.
1. The major characteristics of the standard deviation are:
a. It is in the same units as the original data.
b. It is the square root of the average squared distance from the mean.
c. It cannot be negative.
d. It is the most widely reported measure of dispersion.
2. The formula for the sample standard deviation is
[3–11]
3. The formula for the standard deviation of grouped data is
[3–13]
III. We interpret the standard deviation using two measures.
A. Chebyshev’s theorem states that regardless of the shape of the distribution, at least 1 – 1/k2 of the observations will be within k standard deviations of the mean, where k is greater than 1.
B. The Empirical Rule states that for a bell-shaped distribution about 68 percent of the values will be within one standard deviation of the mean, 95 percent within two, and virtually all within three.
Probability
I. A probability is a value between 0 and 1 inclusive that represents the likelihood a particular event will happen.
A. An experiment is the observation of some activity or the act of taking some measurement.
B. An outcome is a particular result of an experiment.
C. An event is the collection of one or more outcomes of an experiment.
II. There are three definitions of probability.
A. The classical definition applies when there are n equally likely outcomes to an experiment.
B. The empirical definition occurs when the number of times an event happens is divided by the number of observations.
C. A subjective probability is based on whatever information is available.
III. Two events are mutually exclusive if by virtue of one event happening the other cannot happen.
IV. Events are independent if the occurrence of one event does not affect the occurrence of another event.
V. The rules of addition refer to the union of events.
A. The special rule of addition is used when events are mutually exclusive.
B. The general rule of addition is used when the events are not mutually exclusive.
C. The complement rule is used to determine the probability of an event happening by subtracting the probability of the event not happening from 1.
VI. The rules of multiplication refer to the product of events.
A. The special rule of multiplication refers to events that are independent.
B. The general rule of multiplication refers to events that are not i independent.
C. A joint probability is the likelihood that two or more events will happen at the same time.
D. A conditional probability is the likelihood that an event will happen, given that another event has already happened.
E. Bayes’ theorem is a method of revising a probability, given that additional information is obtained. For two mutually exclusive and collectively exhaustive events:
VII. There are three counting rules that are useful in determining the number of outcomes in an experiment.
A. The multiplication rule states that if there are m ways one event can happen and n ways another event can happen, then there are mn ways the two events can happen.
B. A permutation is an arrangement in which the order of the objects selected from a specific pool of objects is important.
C. A combination is an arrangement where the order of the objects selected from a specific pool of objects is not important.
Summary notesExpected values and random variables
I. A random variable is a numerical value determined by the outcome of an experiment.
II. A probability distribution is a listing of all possible outcomes of an experiment and the probability associated with each outcome.
A. A discrete probability distribution can assume only certain values. The main features are:
1. The sum of the probabilities is 1.00.
2. The probability of a particular outcome is between 0.00 and 1.00.
3. The outcomes are mutually exclusive.
B. A continuous distribution can assume an infinite number of values within a specific range.
III. The mean and variance of a probability distribution are computed as follows.
A. The mean is equal to:
B. The variance is equal to:
IV. The binomial distribution has the following characteristics.
A. Each outcome is classified into one of two mutually exclusive categories.
B. The distribution results from a count of the number of successes in a fixed number of trials.
C. The probability of a success remains the same from trial to trial.
D. Each trial is independent.
E. A binomial probability is determined as follows:
F. The mean is computed as:
G. The variance is
V. The hypergeometric distribution has the following characteristics.
A. There are only two possible outcomes.
B. The probability of a success is not the same on each trial.
C. The distribution results from a count of the number of successes in a fixed number of trials.
D. It is used when sampling without replacement from a finite population.
E. A hypergeometric probability is computed from the following equation:
VI. The Poisson distribution has the following characteristics.
A. It describes the number of times some event occurs during a specified interval.
B. The probability of a “success” is proportional to the length of the interval.
C. Nonoverlapping intervals are independent.
D. It is a limiting form of the binomial distribution when n is large and π is small.
E. A Poisson probability is determined from the following equation:
F. The mean and the variance are:
Summary notes probability distributions
I. The uniform distribution is a continuous probability distribution with the following characteristics.
A. It is rectangular in shape.
B. The mean and the median are equal.
C. It is completely described by its minimum value a and its maximum value b.
D. It is also described by the following equation for the region from a to b:
E. The mean and standard deviation of a uniform distribution are computed as follows:
II. The normal probability distribution is a continuous distribution with the following characteristics.
A. It is bell-shaped and has a single peak at the center of the distribution.
B. The distribution is symmetric.
C. It is asymptotic, meaning the curve approaches but never touches the X-axis.
D. It is completely described by its mean and standard deviation.
E. There is a family of normal probability distributions.
1. Another normal probability distribution is created when either the mean or the standard deviation changes.
2. The normal probability distribution is described by the following formula:
III. The standard normal probability distribution is a particular normal distribution.
A. It has a mean of 0 and a standard deviation of 1.
B. Any normal probability distribution can be converted to the standard normal probability distribution by the following formula.
C. By standardizing a normal probability distribution, we can report the distance of a value from the mean in units of the standard deviation.
IV. The normal probability distribution can approximate a binomial distribution under certain conditions.
A. π and n(1 – π) must both be at least 5.
1. n is the number of observations.
2. π is the probability of a success.
B. The four conditions for a binomial probability distribution are:
1. There are only two possible outcomes.
2. π remains the same from trial to trial.
3. The trials are independent.
4. The distribution results from a count of the number of successes in a fixed number of trials.
C. The mean and variance of a binomial distribution are computed as follows:
D. The continuity correction factor of .5 is used to extend the continuous value of X one-half unit in either direction. This correction compensates for approximating a discrete distribution by a continuous distribution.
