QMB 3200 Flashcards
(110 cards)
1
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Data
A
the facts & figures collected, analyzed, and summarized for presentation and interpretation.
2
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Dataset
A
all the data collected for a particular analysis.
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Element
A
the entity on which data is collected.
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Variable
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a characteristic of interest of an element.
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Observation
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the variables associated with an individual element.
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Categorical
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use numeric or ordinal values of measurement of categories.
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Quantitative
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use numeric (quantitative) measures.
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What does statistical analysis depend on
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The type of statistical analysis depends on whether the variable is categorical or quantitative.
9
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Cross Sectional
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data collected at a similar point in time.
10
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Time Series
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data collected over several time periods.
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Panel
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combination of cross-sectional and time series data.
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Descriptive Statistics
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describe data or variables.
13
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Population
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is the set of all data/variables of a statistical analysis.
14
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Sample
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is a subset of the population.
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Statistical Analysis
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uses data from a sample to make estimates and test hypothesis about the characteristics of a population.
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Row
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contains variables names
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Column
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contains the element
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Average Formula
A
=average(A:A)
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Median Formula
A
=median(A:A)
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Analytics
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is the scientific process of transforming data for decision making.
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Descriptive Analytics
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which describe what has happened in the past.
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Predictive Analytics
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uses statistical models frompast data to predict the future [forecasting] or access the impact of one variable on another [inference].
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Prescriptive Analytics
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uses models seeking to find a best (optimal) solution. Often these are some type of optimization model.
24
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Differences between Data and Big Data
A
Volume – the number of observations.
Velocity – the speed at which data is collected.
Variety – the forms of data are of different types.
Veracity – the reliability of the data generated.
*Focus on extracting predictive information from big data
25
Frequency Distribution
a tabular summary of data showing the number (i.e. frequency) of observations in each of several non over-lapping categories.
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Relative Frequency
= frequency of a class/ n of a class
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Percent Frequency
= relative frequency * 100
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Bar Chart
a visual display of frequency; relative frequency & percent frequency distributions. (Used to compare 2 variables)
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Pie Chart
a visual display of frequency; relative frequency & percent frequency distributions.
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A frequency distribution with quantitative data must define the classes for a frequency distribution by:
determine the number of non over-lapping classes;
b. determine the width of each class;
c. determine the class limits.
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Number of Classes
Typically, between 5 and 20.Small datasets have less; larger datasets have more.
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Width of the Class
Generally, it should be the same for each class.
Approximate class width = (largest data value – smallest data value)/number of classes.
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Class Limits
each data observation must only belong to one class.
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Relative Frequency Distributions
= frequency of the class/n.
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Histogram
A visual display of a frequency, relative frequency or percent frequency distribution, where the variable of interest is on the horizontal axis and the frequency, relative frequency or percent frequency is on the vertical axis.
* Shows the shape of the distribution of the variable of interest.
36
Cumulative Distribution
Presents the number of data items with values less than or equal to the upper class limit for each class.
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Cumulative Relative Frequency Distribution
Shows the proportion of data items with values less than or equal to the upper limit of each class.
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Cumulative Percent Frequency Distribution
Shows the percentage of data items with values less than or equal to the upper limit of each class.
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Crosstabulation
a tabular summary of data for two variables (either categorical or quantitative)
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Scatter Diagram & Trendline
A scatter diagram is a graphical display of the relationship between two quantitative variables and a trendline provides an approximation (i.e. an estimate) of the relationship; which can be positive, negative or none.
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Side-by-Side Bar Chart
Depicts multiple bar charts on the same display.
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Stacked Bar Chart
Has one bar broken into segments of a different color showing the relative frequency of each class.
43
Mean(average)
is the average value for a variable and the sample mean is denoted as 𝑥, and the population mean is denoted as 𝜇.
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Sample Mean
∑ 𝑥𝑖/𝑛 ; where Σ is the symbol meaning to sum or add up all the x i ’s. For a variable, the first observation is x 1 , the second is x 2 and the i this x i , and n is the number of observations.
45
Median
is the value in the middle, when the data are arranged in ascending order. When the number of observations are odd the median is the middle value; when the number of observations are even the median is the average of the two middle values. The median avoid problems where there are extreme high or low values for x.
46
Mode
is the value that occurs with the greatest frequency. If there are two values that are most frequent the variable is bi-modal; if there are more then it’s multi-modal.
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Mode Formula
=mode.sngl
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Weighted Mean
used when observations have different weights (relative importance).
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Percentile
provides information about how the data is spread over the interval from the smallest to the largest value. The pth percentile divides the data into two parts – approximately p% of the observations are less than the pth percentile and approx. (100-p)% are greater.
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Location of the pth percentile
𝐿𝑝 = 𝑝 /100 (𝑛 + 1)
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Quartiles
represent how the data is spread over four parts, each containing approximately 25% of the observations. Q 1 is the first quartile (25th percentile); Q2 = second quartile (50th percentile or median); Q3 = third quartile (75th percentile). Same calculation as a percentile, but only use 25th , 50 th and 75 th .
52
Percentile Formula
=percentile.exc( array, k)
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Quartile Formula
=quartile.exc( array, quart )
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Measures of variability or dispersion of the data
Range: largest value – smallest value.Interquartile Range: Q3 – Q1 , is the range of the middle 50% of the data.
Variance: measures variability using all the data, since it is based on the difference between the value of xi and the mean. This difference is called a deviation about the mean. For a sample, a deviation is 𝑥𝑖−̅𝑥 and for a population, a deviation is 𝑥𝑖− 𝜇. Then the deviation is squared.
55
Benefit of Variance
that it is useful in comparing the variability of two or more variables; but one difficulty is the units, since the variance is in units squared.
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Standard Deviation
The population standard deviation: 𝜎 = sqr.rt𝜎 2
The sample standard deviation: 𝑠 = sqr.rt𝑠^2
The advantage is now the units are no longer squared, making it easier to compare the results to the mean as well as other statistics.
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Variance Formula
=var.s (A:A)
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Standard Deviation Formula
=stdev.s (A:A)
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Coefficient of Variation
This is a measure of how large the standard deviation is relative to the mean. Coefficient of Variation = (𝑠/𝑥 ∗ 100)%
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Distribution Shape
is measured by skewness. If the shape of the data is skewed to the left, the skewness is negative; if to the right then skewness is positive; and if the data is symmetric, then skewness is zero.
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Symmetric Distribution
the mean and median are equal.
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Positive Skew Distribution
the mean is usually greater than the median
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Negative Skew Distribution
the mean is usually less than the median.
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Skewness Formula
=SKEW CA,A... A)
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Measure of Relative Location
Measures the relative location of values in the dataset. This helps determine how far a particular value is from the mean.
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Z-Score
The z-Score yields a standardized value and is the number of standard deviations from the mean. The z-Score for any observation is a measure of the relative location of the observation in the dataset.
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Chebyshev's Theorem
Allows us to make statements about the population of the data values that must be within a specified number of standard deviations from the mean. At least (1 − ⁄1 𝑧^ 2) of the data values must be within z standard deviations of the mean (z > 1).
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Chebyshev's Theorem Advantage
It applies to any dataset if the data is bell shaped around the mean
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Detecting Outliers
Outliers are extreme values relative to the rest of the data. z-Score can help identify outliers.Typically, any z-Score greater than 3 is an outlier. Alternatively, we can use the interquartile range, where the:
lower limit: 𝑄1 − 1.5(𝐼𝑄R)
upper limit: 𝑄3 + 1.5(𝐼𝑄R)
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Covariance
is a descriptive measure of the linear association between two variables.
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Covariance Formula
= covariance.s CA:A, B:B) or =covariance.s (array 1, array2)
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Descriptive Measures
Two descriptive measures of the relationship between two variables are covariance and correlation.
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Interpreting Covariance
If sxy > 0, then there is a positive linear association between x and y. If sxy < 0, then there is a negative linear association between x and y.
*Depends on unit of measurement
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Correlation Coefficient Formula
=correl
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Probability
A numerical measure of the likelihood of an event occurring. A probability ranges from 0 to 1, such as the probability it will rain tomorrow.
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Permutations
Is a counting rule computing the number of experimental outcomes when n objects are to be selected from a set of N objects where the order of selections is important.
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Basic Requirements Assigning Probabilities
The probability assigned to each experimental outcome must be between 0 and 1, inclusively.
The sum of the probabilities for all experimental outcomes must be equal to 1.
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3 Methods Assigning Probabilities
Classical Method – such as a coin toss or a roll of a 6-sided die.
Relative Frequency Method - is used when data are available to estimate the proportion of time the experimental outcome will occur if the experiment is repeated a large number of times.
Subjective Method - is used when outcomes are not equally likely and data is unavailable.
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Events
a collection of sample points
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Probability of an Event
is equal to the sum of the probabilities of the sample points in the event.
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Complement of Event A
(A^C) are all the sample points not in the event.
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Union of 2 Events
is the event containing all sample points belonging to Event A, Event B or both. The union of Event A and Event B is denoted by: 𝐴 ∪ 𝐵.
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Intersection of 2 Events
is the event containing the sample points belonging to both A and B. Intersection is denoted by: 𝐴 ∩ 𝐵
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Addition Law
is useful when we want to know the probability that at least one of two events occurs.The addition law: 𝑃 (𝐴 ∪ 𝐵)= 𝑃(𝐴) + 𝑃 (𝐵) −𝑃(𝐴∩𝐵).
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Mutually Exclusive Events
occur when two events have no sample points in common. Addition Law for mutually exclusive events: 𝑃 (𝐴 ∪ 𝐵) = P (A) + P(B)
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Marginal Probabilities
is the sum of the joint probabilities (by row and column).
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Conditional Probability
𝑃 (𝐴|𝐵) = 𝑃(A ∩ 𝐵)/𝑃(𝐵) or𝑃 (𝐵 |𝐴) = 𝑃(𝐴 ∩ 𝐵)/𝑃(𝐴)
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Independent Events
Event A and Event B are independent if:P(A|B) = P(A) or P(B|A) =P(B)
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Multiplication Law
is used to compute the probability of the intersection of two events:𝑃 (𝐴 ∩ 𝐵) = 𝑃 (𝐵) ∗ 𝑃(𝐴|𝐵) or𝑃 (𝐴 ∩ 𝐵) = 𝑃 (𝐴) ∗ 𝑃(𝐵|𝐴)For independent events: 𝑃 (𝐴 ∩ 𝐵) = 𝑃 (𝐴) ∗ 𝑃(𝐵)
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Discrete Random Variables
either are a finite number of values or an infinite number of values such as 0, 1, 2 ...
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Expected Value (or mean)
a random variable is a measure of the central location for a random variable.
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Variance
measures the variability or dispersion of the random variable
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Standard Deviation
is the positive square root of the variance.
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Binomial Probability Distribution
The experiment consists of a sequence of identical trials.
2. Two outcomes are possible on each trial; successor failure.
3. The probability of success (p) and the probability of failure (1-p) does not change from trial to trial.
4. The trials are independent.
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Binomial Distribution Formula
=Binom.Dist
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Poisson Probability Distribution
This distribution relates to the case for estimating the number of occurrences over a specified interval of space/time.
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Properties of a Poisson Experiment
The probability of an occurrence is the same forany two intervals of equal length.
2. Occurrence or non-occurrence in any interval is independent of the occurrence or non-occurrence in any other interval.
In the Poisson distribution, the mean and variance are equal
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Poisson Probability Formula
= Poisson. Dist (x, u, true / False)
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Hypergeometric Probability Distribution
is similar to the binomial distribution, except the trials are not independent and the probability of success changes from trial to trial.
NOTE: r is the number of successes in population N, and N-r is the number of failures.
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Hypergeometric Probability Distribution Formula
= Hypergeom. Dist (x, u, r, N, true/ False)
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Difference Between Discrete and Continuous Random Variables
Discrete random variables are computed where the random variable takes on a specific value; continuous random variables are computed where the random variable is within an interval.
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Exponential Probability Distribution
This distribution is useful for random variable measuring the time between an interval. The exponential probability density function
103
Exponential Probabilities Formula
= Expon. Dist (x, T/M, true / false)
104
Characteristics of the Normal Distribution
Only two parameters: μ and σ.
2. The highest point is the mean, which is also the median and the mode.
3. The mean can take on any numerical value.
4. The normal distribution is symmetric; skewness = 0
The standard deviation determines how flat or wide the normal curve is. Larger standard deviations result in wider or flatter curves.
6. Probabilities for a normal random variable are given by the area under the normal curve. Total area under the curve equals 1.
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Normal Distribution Formula
=Norm. Dist (x,u, o, true, false)
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Inverse Norm Formula
=Norm.Inv
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Standard Normal Probability Distribution
Standard Normal Probability Distribution is where the μ is 0 and the standard deviation is 1.
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Standard Normal Distribution Formula
= Norm. s. Dist (z, true)
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Outliers Formula
= quartile. exe (array, quart)
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Poisson Probabilities Formula
= Poisson. Dist (x, u, true / False)
