statistics
the methodology of extracting useful information from a data set
find the right data
use appropriate statistical tools
clearly communicate the numerical information into written language
to do good statistics...
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descriptive statistics & inferential statistics
two branches of statistics
descriptive statistics
collecting, organizing, and presenting the data
inferential statistics
drawing conclusions about a population based on sample data from that population
population
consists of all items of interest
sample
a subset of the population
sample statistic
calculated from the data and is used to make inferences about the population parameter
too expensive to gather information on the entire population
often impossible to gather information on the entire population
reasons for sampling from the population
cross-sectional data
data collected by recording a characteristic of many subjects at the same point in time, or without regard to differences in time
individuals
households
firms
industries
regions
countries
subjects of cross-sectional data
time series data
data collected by recording a characteristic of a subject over several time periods
daily
weekly
monthly
quarterly
annual
(observations)
time series data can include:
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35 is likely the estimated average age. It would be rather impossible to reach all actual video game players
Many people regard video games as an obsession for youngsters, but in fact, the average age of video game players is 35 years. Is the value likely the actual or estimated average age of the population? Explain.
All marketing managers
No! The average salary was likely computed from a sample in order to save time and money.
Business graduates in the U.S. with a marketing concentration earn high salaries. According to the Bureau of Labor Statistics, the average annual salary for marketing managers was $104,400 in 2007.
What is the relevant population?
Do you think the average
variable
general characteristic being observed on an object of interest
qualitative & quantitative
types of variables
gender
race
political affiliation
(categorical)
qualitative variables examples
test scores
age
weights
quantitative variables examples
discrete
assumes a countable number of distinct values
discrete & continuous
types of quantitative variables
continuous
can assume an infinite number of values within some interval
discrete
The number of children in a family, or the number of points scored in a basketball game is an example of what quantitative variable?
continuous
weight, height, and investment return are examples of what quantitative variable?
nominal & ordinal
qualitative scales of measurement
interval & ratio
quantitative scales of measurement
nominal scale
least sophisticated level of measurement; data are simply categories for grouping data
ordinal scale
data may be categorized AND ranked with respect to some characteristic or trait
differences between categories are meaningless because the actual numbers used may be arbitrary
interval scale
data may be categorized AND ranked with respect to some characteristic or trait
differences between the values are equal and meaningful. Thus arithmetic operations of addition and subtraction are meaningful.
No "absolute 0" or starting point defined. Mean
ratio scale
strongest level of measurement
data may be categorized AND ranked with respect to some characteristic or trait
differences between interval values are equal and meaningful.
there is an "absolute 0" or defined starting point. "0" does mean "the absence of.
weight...sales
time...profits
distance...inventory levels
these variables are measure on a ratio scale
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qualitative
is the following variable qualitative or quantitative?
if quantitative, is it discrete or continuous?
Colors of cars in a mall parking lot:
quantitative and continuous
is the following variable qualitative or quantitative?
if quantitative, is it discrete or continuous?
Time it takes each student to complete a final exam:
quantitative and discrete
is the following variable qualitative or quantitative?
if quantitative, is it discrete or continuous?
The number of patrons who frequent a restaurant
quantitative and ratio
define the type of measurement scale:
an investor collects data on the weekly closing price of gold throughout a year
qualitative and ordinal
define the type of measurement scale:
analyst assigns a sample of bond issues to one of the following credit ratings: AAA, AA, BBB, BB, CC, DD
qualitative and nominal
define the type of measurement scale:
the dean of the business school categorizes the students by majors (i.e. accounting, marketing, management, isds, economics, finance)
true
true or false?
a statistic is usually unobservable while a parameter is usually observable.
true
true or false?
a sample is the portion of the universe that is selected for analysis
inferential
Mediterranean fruit flies were discovered in CA a few years ago and badly damaged the oranges grown in that state. Suppose the manager of a large farm wanted to study the impact of the fruit flies on the orange crops on a daily basis over a 6-week period.
descriptive
The Human Resources Director of a large corporation wishes to develop an employee benefits package and decides to select 500 employees from a list of all (N=40,000) workers in order to study their preferences for the various components of a potential pack
2,100 business college students at the time of the study
statistics
96 students involved in the study
Three professors at NKU compared two different approaches to teaching courses in the school of business. At the time of the study, there were 2,100 students in the business school and 96 students were involved in the study. Demographic data collected on t
parameter
Mediterranean fruit flies were discovered in CA a few years ago and badly damaged the oranges grown in that state. Suppose the manager of a large farm wanted to study the impact of the fruit flies on the orange crops on a daily basis over a 6-week period.
inferential statistics
The Human Resources Director of a large corporation wishes to develop an employee benefits package and decides to select 500 employees from a list of all (N=40,000) workers in order to study their preferences for the various components of a potential pack
true
true or false?
Compiling the number of registered voters who turned out to vote for the primary in Iowa is an example of descriptive statistics.
descriptive
The Commissioner of Health in NY State wanted to study malpractice litigation in NY. A sample of 31 thousand medical records was drawn from a population of 2.7 million patients who were discharged during the year 2009. The collection, presentation, and ch
statistics
Mediterranean fruit flies were discovered in CA a few years ago and badly damaged the oranges grown in that state. Suppose the manager of a large farm wanted to study the impact of the fruit flies on the orange crops on a daily basis over a 6-week period.
parameters
The Quality Assurance Department of a large urban hospital is attempting to monitor and evaluate patient satisfaction with hospital services. Prior to discharge, a random sample of patients is asked to fill out a questionnaire to rate such services as med
variable
a characteristic of an item or individual
operational definitions
universally accepted meanings that are clear to all associated with an analysis
statistic
a measure that describes a characteristic of a sample
population
all the items or individuals about which you want to draw a conclusion
inferential statistics
methods that use the data collected from a small group to draw conclusions about a larger group
parameter
a measure that describes a characteristic of a population
data
different values associated with a variable
sample
the portion of a population selected for analysis
descriptive statistics
methods that help collect, summarize, present, and analyze a set of data
true
true or false?
A professor computed the sample average exam score of 20 students and used it to estimate the average exam score of the 1,500 students taking the exam was an example of inferential statistics.
frequency distribution
for qualitative data groups data into categories and records how many observations fall into each category
relative frequency
divide each category's frequency by the sample size
100
to express relative frequencies in terms of percentages, multiply each proportion by ______
1.00 & 100%
Note that with relative frequency, the total of the proportions must add to be ___ and the total of the percentages must add to be _____
pie chart
is a segmented circle whose segments portray the relative frequencies of the categories of some qualitative variable
bar chart
depicts the frequency or the relative frequency for each category of the qualitative data as a bar rising vertically from the horizontal axis
frequency distribution
for quantitative data groups data into intervals called classes, and records the number of observations that fall into each class
classes are mutually exclusive
classes are exhaustive
guidelines when constructing frequency distribution
5-20
the number of classes usually ranges from __ to __
largest value-smallest value
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number of classes
approximating the class width:
cumulative frequency distribution
specifies how many observations fall below the upper limit of a particular class
relative frequency distribution
identifies the proportion or fraction of values that fall into each class
class frequency
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total number of observations
class relative frequency =
cumulative relative frequency distribution
gives the proportion or fraction values that fall below the upper limit of each class
histogram
a visual representation of a frequency or a relative frequency distribution
respective class frequency (relative frequency)
bar height in a histogram represents
class width
bar width in a histogram represents
y-axis scale
the only difference between frequency and relative frequency histograms is the
shape of distribution
typically symmetric or skewed
symmetric
mirror image on both sides of its center
positively skewed
data form a long, narrow tail to the right
negatively skewed
data form a long, narrow tail to the left
polygon
is a visual representation of a frequency or a relative frequency distribution
x-axis, y-axis
In a polygon:
plot the class midpoints on the ______ and associated frequency or (relative frequency) on the ______
straight line
neighboring points in a polygon are connected with a _______ ______
ogive
is a visual representation of cumulative frequency or a cumulative relative frequency distribution
stem-and-leaf diagram
provides a visual display of quantitative data; gives an overall picture of the data's center and variability
stem...leaf
each value in a stem-and-leaf data set is separated into two parts: the _____ consists of the leftmost digits, and the ____ is the last digit
scatterplot
is used to determine if two variables are related
linear relationship
upward or downward sloping trend of data
positive
linear relationship in which as x increases, so does y
negative
linear relationship in which as x increases, y decreases
curvilinear relationship
linear relationship in which as x increases, y increases at an increasing (or decreasing) rate
as x increases, y decreases at an increasing (or decreasing) rate
no relationship
linear relationship in which data are randomly scattered with no discernible pattern
true
true or false?
the quality ("terrible", "poor", "fair", "acceptable", "very good", and "excellent") of a day care center is an example of a nominal scaled variable.
False. Ogive is from cumulative %. Polygon
If you wish to construct a graph of a relative frequency distribution, you would most likely construct an ogive.
false. frequency
true or false?
the sum of cumulative frequencies in a distribution always equals 1
true
true or false?
the amount of calories contained in a pack of 12-ounce cheese is an example of a discrete variable
false
true or false?
faculty rank (professor or lecturer) is an example of discrete numerical data
false
true or false?
the grade level (K-12) of a student is an example of a numerical variable
discrete
An insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. The number of claims a person has made in the last 3 years is an example of a ________ numerical variable
discrete
A personal computer user survey was conducted. Number of personal computers owned is an example of a ________ numerical variable
true
true or false?
the relative frequency is the frequency in each class divided by the total number of observations
continuous
a personal computer user survey was conducted, hours of personal computer use per week is an example of a ___________ numerical variable
categorical
in purchasing an automobile, there are a number of variables to consider... the color of the car is an example of a ___________ variable
continuous
An insurance company evaluates many numerical variables about a person before deciding on an appropriate rate for automobile insurance. The distance a person drives in a year is an example of a __________ numerical variable
true
true or false?
the possible responses to the question "how long have you been living at your current residence?" are values from a continuous variable
false
true or false?
the amount of calories contained in a pack of 12-ounce cheese will be measured on a ratio scale.
class midpoint
the point halfway between the boundaries of each class interval in a grouped frequency distribution is called the _______________
false... usually between 5 and 15
true or false?
In general, a frequency distribution should have at least 8 class groups but no more than 20
true
true or false?
the possible responses to the question "how many times in the past three months have you visited a city park?" are values from a discrete variable.
true
true or false?
the larger the number of observations in a numerical data set, the larger the number of class intervals needed for a grouped frequency distribution
categorical
in purchasing an automobile, there are a number of variables to consider; the body style of the car (sedan, coupe, wagon, etc.) is an example of a ___________ variable
false... use frequency polygon
true or false?
apple computer inc. collected information on the age of their customers; the youngest customer was 12 and the oldest was 72; to study the distribution age among its customers, it can use a percentage polygon
false
true or false?
the number of defective apples in a single box is an example of a continuous variable
fals... in a histogram there are no gaps between adjacent bars
true or false?
a histogram can have gaps between the bars, whereas bar charts cannot have gaps
central tendency
the extent to which all the data values group around a typical or central value
variation
the amount of dispersion or scattering of values
shape
the pattern of the distribution of values from the lowest value to the highest value
mean
most common measure of central tendency; sum of values divided by the number of values; affected by extreme values (outliers)
median
in an ordered array, the ______ is the "middle" number (50% above, 50% below); not affected by extreme values
n+1
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2
median position=
position in the ordered data, not the value of the median
rules of the median position
if the number of values is odd, the median is the middle number
if the number of values is even, the median is the average of the two middle numbers
mode
value that occurs most often; not affected by extreme values; used for either numerical or categorical (nominal) data; there may be none, but there may be several
mean
which measure to choose?
is generally used, unless extreme values (outliers) exist
median
which measure to choose?
is often used, since it is not sensitive to extreme values
mean and median
which measure to choose?
in some situations it makes sense to report on ____ and ______
measures of variation
give information on the spread of variability or dispersion of the data values
range
simplest measure of variation; difference between the largest and the smallest values; sensitive to outliers
ignores the way in which data are distributed
why can the range be misleading?
sample variance
average (approximately) of squared deviations of values from the mean
sample standard deviation
most commonly used measure of variation; shows variation about the mean; is the square root of the variance; has the same units as the original data
steps for computing sample standard deviation
1. compute the difference between each value and the mean
2. square each difference
3. add the squared differences
4. divide this total by n-1 to get the sample variance
5. take the square root of the sample variance to get the sample standard deviation
greater
the more the data are spread out, the _______ the range, variance, and standard deviation
smaller
the more the data are concentrated, the _______ the range, variance, and standard deviation
zero
if all the values are the same (no variation), all these measures will be ____
negative
range, variance, and standard deviation are never ________
z-score
to compute the _______ of a data value, subtract the mean and divide by the standard deviation
z-score
is the number of standard deviations a data value is from the mean
extreme outlier
a data value is considered an _______ _______ if its Z-score is less than -3.0 or greater than +3.0
larger...farther
the ______ the absolute value of the Z-score, the _______ the data value is from the mean
x-X| data value-sample mean
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S sample standard deviation
Z=
z-score formula
shape of a distribution
describes how data are distributed
skewness and kurtosis
two useful shape related statistics are:
skewness
measures the amount of asymmetry in a distribution
kurtosis
measures the relative concentration of values in the center of a distribution as compared with the tails
left skewed
long tail (distortion) to the left caused by extremely low values, pulls down the mean so it is less than the median
mean<median
symmetric
right and left tails are equal so mean=median
right skewed
long tail (distortion) to the right caused by extremely high values which pull the mean upward so mean is greater than median
mean > median
quartiles
five-numbered summary
boxplot
another way to describe numerical data
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quartile measures
split the ranked data into 4 segments with an equal number of values per segment
Q1=(n+1)/4
Q2=(n+1)/2
Q3=3(n+1)/4
where n is the number of observed values (positions)
find a quartile by determining the value in the appropriate position in the ranked data
3 formulas:
quartile measure: calculation rules
1. if the result is a whole number then it is the ranked position to use
2. if the result is a fractional half (2.5,7.5,8.5, etc.) then average the two corresponding data values
3. if the result is neither (not whole or fractional half) then round the res
median
Q2 is a measure of central tendency or ______
IQR- interquartile range
is Q3-Q1 and measures the spread in the middle 50% of the data
midspread
IQR is also called the _________ because is covers the middle 50% of the data
influenced by outliers or extreme values
IQR is a measure of variability that is not __________ __ ________ __ _______ ______
resistant measures
measures like the median, Q1, Q3, and IQR that are not influenced by outliers are called _________ ________
the five-number summary
the five numbers that help describe the center, spread, and shape of the data
boxplot
a graphical display of the data based on the five-number summary; implies a skewness to the right
median
in a boxplot:
vertical line inside the box
location of Q1
in a boxplot:
vertical line at the left side of box is the
location of Q3
in a boxplot:
vertical line at right side of box is the
smalles value
the vertical line at the far left is the
25% of the data
area between the left side of the box and the smallest value represents
middle fifty
the area inside the box is the
largest value
vertical line to the far right is the
the top 25% of the data
area between the right side of the box and the largest value represents
symmetrical data
the box plot implies
median...endpoints
if data are symmetric around the ______ then the box and central line are centered between the _________
vertical or horizontal
a boxplot can be shown in either a ________ or __________ orientation
sample...population
descriptive statistics discussed previously described a ______ not the __________
parameters
summary measures describing a population are called __________, and are denoted with Greek letters
population mean
population variance
population standard deviation
important population parameters are
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population mean
is the sum of the values in the population divided by the population size
population variance
average of squared deviations of values from the mean
population standard deviation
most commonly used measure of variation; shows variation about the mean; is the square root of the population variance; has the same units as the original data
empirical rule
approximates the variation of data in a bell-shaped distribution
68%
about ___ of the data in a bell shaped distribution is within 1 standard deviation of the mean
95%
about ___ of the data in a bell-shaped distribution lies within two standard deviations of the mean
99.7%
about _____ of the data in a bell-shaped distribution lies within three standard deviations of the mean
empirical rule
helps measure how values distribute above and below the mean and help identify outliers
data analysis is objective
should report the summary measures that best describe and communicate the important aspects of the data set
data interpretation is subjective
should be done in fair, neutral and clear manner
ethical considerations
numerical descriptive measures:
-should document both good and bad results
-should be presented in a fair, objective and neutral manner
-should not use inappropriate summary measures to distort facts
20
The data below represent the amount of grams of carbohydrates in a serving of breakfast cereal in a sample of 11 different servings
11 15 23 29 19 22 21 20 15 25 17
The median carbohydrate amount in the cereal is __ grams.
895.5
the number of values is even, so the median is the average of the two middle numbers
869+922=1791/2=895.5
what is the median of the set of data
308 423 593 708 869 922 1223 1425 1589 1720
true
true or false?
a boxplot is a graphical representation of a five-number summary
false
true or false?
the median of the values 3.4, 4.7, 1.9, 7.6, and 6.5 is 4.05; put the array in order. median is the middle value
1.9, 3.4, 4.7, 6.5, 7.6
true
true or false?
as a general rule, an observation is considered an extreme value if its Z-score is greater than 3
-3.0...+3.0
a data value is considered an extreme outlier if its z-score is less than ____ or greater than ____
25%
only 25% of the observations are greater than the third quartile
you were told that the 1st, 2nd, and 3rd quartiles of students' weight at a major university are 95 lbs, 125 lbs, and 138 lbs; what percentage of the student weigh more than 138 lbs?
false
Q1 and Q3 are measures of non-central location
Q2= the median, a measure of central tendency
true or false?
the interquartile range is a measure of central tendency in a set of data
true
*if the number value is even, the median is the average of the two middle numbers
true or false?
the median of a data set with 20 items would be the average of the 10th and the 11th items in the ordered array
false
true or false?
the larger the Z-score, the farther is the distance from the observation to the median
true
true or false?
the z-scores can be used to identify outliers
true
true or false?
in exploratory data analysis, a box plot can be used to illustrate the median, quartiles, and extreme values
true
true or false?
the line drawn within the box of the boxplot always represents the median
mean absolute deviation
average of the absolute differences between the values of the data set and the mean
arithmetic mean
the additive average of the values in the data set
geometric mean
the multiplicative average of the values in a data set
geometric
the _________ mean is the appropriate measure to use when evaluating growth rates
A
the greater the coefficient, the greater the dispersion
if fund A has a coefficient of variation of 1.1, and fund B has a coefficient of variation of 0.9, fund _ has the greater relative dispersion because...________________
weighted mean
where a mean is calculated and some observations are given greater importance or value, the mean is known as a ________ mean
outliers
________ may distort the mean as a measure of central location
median
when outliers are present the ______ is the best measure of central location
mode
when summarizing a qualitative data set the ____ is the best measure of central location
dividing the standard deviation by the mean
o/mu
the coefficient of variation can be found by: