# Glossary

95% Rule
On a normal distribution approximately 95% of data will fall within two standard deviations of the mean; this is an abbreviated form of the Empirical Rule
Alternative Hypothesis
The statement that there is some difference in the population(s), denoted as $$H_a$$ or $$H_1$$
Association
A relationship between variables
Bar chart
Graphical representation for categorical data in which vertical (or sometimes horizontal) bars are used to depict the number of experimental units in each category; bars are separated by space. Penn State Fall 2017 Undergraduate Enrollments

Bias
The systematic favoring of certain outcomes.
Binomial random variable
A specific type of discrete random variable that counts how often a particular event occurs in a fixed number of tries or trials.
Blinding
Procedure employed in research to prevent bias in which the participants and/or the researchers interacting with the participations do not know which treatment each case is receiving.
Bootstrapping
A resampling procedure for constructing a sampling distribution using data from a sample.
Case
An experimental unit from which data are collected
Categorical variable
Names or labels (i.e., categories) with no logical order or with a logical order but inconsistent differences between groups, also known as qualitative.
Causation
Changes in one variable can be attributed to changes in a second variable.
Clustered Bar Chart
Each bar represents one combination of the two categorical variables (i.e., one cell in a contingency table). This is also known as a side-by-side bar chart. Complement

The probability that the event does not occur. The complement of $$P(A)$$ is $$P(A^C)$$. This may also be written as $$P(A')$$.

In the diagram below we can see that $$A^{C}$$ is everything in the sample space that is not A.

Conditional Probability

The probability of one event occurring given that it is known that a second event has occurred. This is communicated using the symbol $$\mid$$ which is read as "given."

For example, P(A\mid B) is read as "Probability of A given B."

Confidence Interval
A range computed using sample statistics to estimate an unknown population parameter with a stated level of confidence.
Confounding Variable
Characteristic that varies between cases and is related to both the explanatory and response variables; also known as a lurking variable or a third variable.
Continuous variable
Characteristic that varies and can take on any value and any value between values.
Control Group
A level of the explanatory variable that does not receive an active treatment; they may receive no treatment or a placebo.
Convenience Sampling
A method of obtaining a sample from a population by ease of accessibility; such a sample is not random and may not be representative of the intended population.
Correlation
A measure of the direction and strength of the relationship between two variables.
Deviation
An individual score minus the mean.
Discrete variable
Characteristic that varies and can only take on a set number of values.
Disjoint Events

Two events that do not occur at the same time. These are also known as mutually exclusive events

In the Venn diagram below event A and event B are disjoint events because the two do not overlap.

Dotplot Double-Blind Study
Research study in which neither the participants nor the researchers interacting with them know which cases have been assigned to which treatment groups.
Empirical Rule
On a normal distribution about 68% of data will be within one standard deviation of the mean, about 95% will be within two standard deviations of the mean, and about 99.7% will be within three standard deviations of the mean. Experimental Research Design
A study in which the researcher manipulates the treatments received by subjects and collects data; also known as a scientific study
Explanatory Variable
Variable that is used to explain variability in the response variable, also known as an independent or predictor variable, it explains variations in the response variable; in an experimental study, it is manipulated by the researcher.
Frequency Table
A table containing the counts of how often each category occurs.
Summary Statistics
Campus Count Percent
University Park 40835 48.5%
Commonwealth Campuses 29388 34.9%
PA College of Technology 5465 6.5%
World Campus 8513 10.1%
Total 84201 100.0%

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Histogram Independent Events
Unrelated events. The outcome of one event does not impact the outcome of the other event.
Independent Groups
Cases in each group are unrelated to one another.
Inferential Statistics
Statistical procedures that use data from an observed sample to make a conclusion about a population.
Interquartile range (IQR)
The difference between the first and third quartiles.
Intersection

The overlap of two or more events and is symbolized by the character $$\cap$$.

$$P(A \cap B)$$ is read as "the probability of A and B."

Least squares method
Method of constructing a regression line which makes the sum of squared residuals as small as possible for the given data.
Left Skewed
A distribution in which the lower values (towards the left on a number line) are more spread out than the higher values. This is also known as negatively skewed. Margin of Error
Half of the width of a confidence interval; equal to the multiplier times the standard error.
Mean

The numerical average; calculated as the sum of all of the data values divided by the number of values.

The sample mean is represented as $$\overline{x}$$ ("x-bar") and the population mean is denoted as the Greek letter $$\mu$$ ("mu"). The formula is the same for the sample mean and the population mean.

Median
The middle of the distribution that has been ordered from smallest to largest; for distributions with an even number of values, this is the mean of the two middle values.
Mode
The most frequently occurring value(s) in the distribution, may be used with quantitative or categorical variables.
Non-Response Bias
Systematic favoring of certain outcomes that occurs when the individuals who choose participate in a study differ from the individuals who choose to not participate.
Normal Distribution
One specific type of symmetrical distribution. This is also known as a bell-shaped distribution. Null Hypothesis
The statement that there is not a difference in the population(s), denoted as $$H_0$$
Observational Research Design
A study in which the researcher collects data without performing any manipulations; also known as a non-experimental study
Odds
Express risk by comparing the likelihood of an event happening to the likelihood it does not happen. Note that the interpretation of odds is different from the interpretation of risk/probability/proportion.
p-value
Given that the null hypothesis is true, the probability of obtaining a sample statistic as extreme or more extreme than the one in the observed sample, in the direction of the alternative hypothesis.
Paired Groups
Cases in each group are meaningfully matched with one another; also known as dependent samples or matched pairs.
Parameter
A measure concerning a population (e.g., population mean).
Percentile
Proportion of a distribution less than a given value.
Pie chart

Graphical representation for categorical data in which a circle is partitioned into “slices” on the basis of the proportions of each category.

Placebo Group
A group that receives what, to them, appears to be a treatment, but actually is neutral and does not contain any active treatment (e.g., a sugar pill in a medication study).
Point Estimate
Sample statistic that serves as the best estimate for a population parameter.
Population
The entire set of possible cases.
Quantitative variable
Numerical values with magnitudes that can be placed in a meaningful order with consistent intervals, also known as numerical.
Randomization
The act of randomly assigning cases to different levels of the explanatory variable.
Range
The difference between the maximum and minimum values.
Relative Risk
Relative risk compares the risk of a particular outcome in two different groups.
Representative Sample
A subset of the population from which data are collected that accurately reflects the population.
Residual
The difference between an observed y value and the predicted y value. In other words, $$y-\widehat y$$. On a scatterplot, this is the vertical distance between the line of best fit and the observation. In a sample this may be denoted as $$e$$ or $$\widehat \epsilon$$ ("epsilon-hat") and in a population this may be denoted as $$\epsilon$$ ("epsilon").
Response Bias
Systematic favoring of certain outcomes that occurs when participants do not respond truthfully; they may do so to align with social norms or to appease the researcher.
Response Variable
Also known as the dependent or outcome variable, its value is predicted or its variation is explained by the explanatory variable; in an experimental study, this is the outcome that is measured following manipulation of the explanatory variable.
Right Skew
A distribution in which the higher values (towards the right on a number line) are more spread out than the lower values. This is also known as positively skewed. Risk
The probability that an event will occur. It may be written as a decimal, a fraction, or a percent.
Sample
A subset of the population from which data are collected,
Sampling Bias
Systematic favoring of certain outcomes due to the methods employed to obtain the sample.
Sampling Distribution
Distribution of sample statistics with a mean approximately equal to the mean in the original distribution and a standard deviation known as the standard error.
Scatterplot
A graphical representation of two quantitative variables in which the explanatory variable is on the x-axis and the response variable is on the y-axis.
Segmented Bar Chart
Also known as a stacked bar chart, one categorical variable is represented on the x-axis while the second categorical variable is denoted within the bars. Minitab Express will not construct a stacked bar chart, but other softwares will. The segmented bar chart below was constructed using Excel. Simple Random Sampling
A method of obtaining a sample from a population in which every member of the population has an equal chance of being selected.
Single Boxplot
Graph displaying data from one quantitative variable. Also known as a "box-and-whisker plot." The box represents the middle 50% of observed values. The bottom of the box is the first quartile (25th percentile) and the top of the box is the third quartile (75th percentile). The line in the middle of the box is the median (50th percentile). The lines, also known as whiskers, extend to the lowest and highest values that are not outliers. Outliers are symbolized using asterisks or circles.
Single-Blind Study
Research study in which the participants do not know the treatment group that they have been assigned to.
Skewed
A distribution in which values are more spread out on one side of the center than on the other.
Standard Deviation
Roughly the average difference between individual data values and the mean. The standard deviation of a sample is denoted as $$s$$. The standard deviation of a population is denoted as $$\sigma$$.
Standard Error
Standard deviation of a sampling distribution.
Statistic
A measure concerning a sample (e.g., sample mean).
Statistical literacy

“People’s ability to interpret and critically evaluate statistical information and data-based arguments appearing in diverse media channels, and their ability to discuss their opinions regarding such statistical information” (Gal, as cited by Rumsey, 2002)

Rumsey, D. J. (2002). Statistical literacy as a goal for introductory statistics courses. Journal of Statistics Education, 10(3). Retrieved from: http://www.amstat.org/publications/jse/v10n3/rumsey2.html

Statistical significance
Sample statistics vary from the specified population parameters to the extent that it is unlikely that the results obtained were due to random sampling error, rather we conclude that the differences observed in the sample were due to actual differences in the population.
Statistics
The art and science of answering questions and exploring ideas through the processes of gathering data, describing data, and making generalizations about a population on the basis of a smaller sample.
Sum of Squared Deviations
Deviations squared and added together. This is also known as the sum of squares or SS.
Sum of squared Residuals
The sum of all of the residuals squared: $$\sum (y-\widehat{y})^2$$.
Symmetrical Distribution
A distribution that is similar on both sides of the center. Two-Way Table
A display of counts for two categorical variables in which the rows represent one variable and the columns represent a second variable. Also known as a contingency table.
Type I Error
Rejecting $$H_0$$ when $$H_0$$ is really true, denoted by $$\alpha$$ ("alpha") and commonly set at .05.
$$\alpha=P(Type\;I\;error)$$
Type II Error
Failing to reject $$H_0$$ when $$H_0$$ is really false, denoted by $$\beta$$ ("beta").
$$\beta=P(Type\;II\;error)$$
Union

A union contains the area in A or B and is symbolized by $$\cup$$. Note that this also includes the overlap of A and B (i.e., the intersection).

$$P(A \cup B)$$ is read as "the probability of A or B."

Variable
Characteristic of cases that can take on different values (in other words, something that can vary).
Variance
Approximately the average of all of the squared deviations; for a sample represented as $$s^{2}$$.
Venn diagram
A visual representation in which the sample space is depicted as a box and events are represented as circles within the sample space.
z-distribution
A bell-shaped distribution with a mean of 0 and standard deviation of 1, also known as the standard normal distribution. z-score
Distance between an individual score and the mean in standard deviation units; also known as a standardized score.