STAT201 Midterm

Observation

A single member of a collection of items that we want to study, such as a person, firm, or region

Variable

A characteristic of the subject or individual, such as an employee’s income or an invoice amount

Data Set

Consists of all the values of all of the variables for all of the observations we have chosen to observe

Categorical/Qualitative Data

Items that are put into different, labeled boxes

Verbal Categorical/Qualitative Data

Items that are put into labeled boxes with no numeric meaning

Coded Categorical/QualitativeData

Items that are put into labeled boxes with numeric meaning, typically sequential

Numerical/Quantative Data

Items that make up a dataset are numbers

Discrete Numerical/Quantative Data

Items that make up a dataset are integers (whole) numbers

Continuous Discrete Numerical/Quantative Data

Items that make up a dataset are numbers that can have decimal places (even if none in the data set have any)

Time Series Data

Each observation in the sample represents a different equally spaced periods

Periodicity

The time between observations

Cross-Sectional Data

Each observation represents a different individual unit at the same point in time

Nominal Data

Not in a natural ordering, no meaning scale distances, and no meningful zero. Identifies a category

Ordinal Data

In a natural ordering, no meaning scale distances, and no meningful zero. Can be ranked.

Interval Data

In a natural ordering, has meaning scale distances, but no meningful zero. Can be ranked.

Ratio Data

In a natural ordering, has meaning scale distances, and has a meningful zero. Can be ranked.

Likert Scales

Interval data frequently used in survey research. Options range from 5-7, 1 to 5 or -2 to 2

Population

All items that are could be part of the data set

Sample

A subset of items that actually make up the data set

Census

An examination of all items in a defined population

Parameter

A measurement of characteristic of the population

A measurement of characteristic of the Sample

μ/mu

Greek symbol used to refer to mean. Alternative when there are multiple means to be calcukated is x ̅/x bar

π/pi

Greak symbol used to refer to population proportion. Alternative when there are multiple population proportions to be calculated is p

Target Population

All possible items that we could be interested in

Sampling Frame

The group we use to represent the population

Systematic Sample

Slecint every nth item from a list or sequence

Stratified Sample

Select randomly within defined conditions

Cluster Sample

Select random geographical regions to represent the population

Simple Random Sample

Every item in the population of N items has the same chance of being chosen

Replacement

The concept of putting data back into a data set after it has been measured so that it has the possibility of being randomly chosen again

Strata

Relatively homogeneous subgroups of known size

One-Stage Cluster Sampling

Sample consists of all elements in each of n randomly chosen subregions

Two-Stage Cluster Sampling

Choose n subregions, then choose a random sample of elements within each cluster

Judegment Sample

Use expert knowledge to choose “typical” items

Convenience Sample

Use a sample that happens to be available

Focus Groups

In-depth dialog with a representative panel of individuals

Qouta Sampling

Use expert knowledge to choose “typical” items while ensuring a certain amount of items with certain qualities are present

Sample Size

Depends on the inherent variability of the quantity being measured and the desired precision of the estimate. High variability needs more data to get an accurate outcome

Bias

A systematic tendency to over- or underestimate a population parameter of interest

Error

Generally refers to problems in sample methodology that lead to inaccurate estimates of a population parameter

Nonresponse Bias

Respondents differ from non-respondents

Selection Bias

Seld-selected respondents are atypical

Response Error

Respondents give false information

Coverage Error

Incorrect specification of frame or population

Measurement Error

Unclear survery instrument wording

Interviewer Error

Responses influenced by interviewer

Sampling Error

Random and unavoidable

Visual Data

Provides insight into characteristics of a data set without using front-facing mathematics

Numerical Data

Provides insight into characteristics of a data set using mathematics

Measurement

The units items are being counted in

Center

Where the data in a graph is conentrated

Variability

The amount data changes in the sample

Shape

How the data physically positioned on the graph

Stem-And-Leaf Plots

A frequency tally that uses digits instead of tally marks to reveal essential data features in intuitive ways. The stem is the tens digit and beyond, the lead is the ones digit

Dot Plots

The simplest graphical display of n individual values of numerical data, revealing dispersion, central tendency, adn the shape of the distribution in an easy to understand way

Histograms

A graphical representation of a frequency distribution. The Y-axis is the frequency within each bin, and the x-axis ticks show the ends points of each bin

Skewness

The direction of the longer tail of the histogram. (n/((n-1)*(n-2)))*(_i=1^kΣ(x_i-[x bar])/s)³)

Left/Negatively Skewed

There is less data on the left side of the histogram. Mean < Median or Median > Midhinge

Right/Positively Skewed

There is less data on the right side of the histogram. Mean > Median or Median < Midhinge

Symmetric

Neither tail of the histogram is unreasonably longer than the other. Mean ≈ Median or Median ≈ Midhinge

Outlier

An extreme value that is far enough from the majority of the data that it probably arose from a different cause or is due to measurement error. Data beyond three standard deviations from the mean. Will lay outside the outer fences of a box plot

Frequency Polygon

A line graph that connects the midpoints of the histogram intervals, plus extra intervals at the beginning and end so that the line will touch the X-axis

Ogive

A line graph of the cumulative frequencies

Arithmetic Scale

Distances on the Y-axis are proportional to the magnitude of the variable being displayed on a log scale

Logarithmic Scale

Equal distances represent equal ratios on a log scale

Log Scale

Y-axis measurements that help visualize how data changes on a line graph

Scatter Plots

A bunch of dots on a chart that can convet patterns in the data pair

(Strong/Weak) Positive Correlation

As Y/X increases, X/Y also Increases

(Strong/Weak) Negative Correlation

As Y/X increases, X/Y decreases

No Pattern

As Y/X increases, X/Y has no noticable trend with it

Nonlinear Pattern

As Y/X increases, X/Y has a patternistic, but not consistent, reaction

Table

Arragement of data into rows and columns

Mean/Sigma/Σ

The average of the data. (_i=1ⁿΣ)/n

Median

The exact middle of sorted data. Q₂/Second Quartile cut-off

Mode

The most commonly occuring value in the data. If multiple numbers tie, then they are both this

Midrange

Halfway between the lowest and highest values of X. (x_min+x_max)/2

Geometric Mean/G

ⁿroot(x₁*x₂*…*x_n)

Trimmed Mean

The average of the data after taking n% off both sides (not total off)

Growth Rate

^n-1root(x_n/x₁)-1

Variation

The spread of the data points around the center of the distribution

Range

The difference between the largest and smallest observations. x_max-x_min

Variance/s²/Var(X)

For Sample: (_i=1ⁿΣ(x₁-[x bar]) ²)/n-1 For Population: (_i=1ⁿΣ(x₁-[x bar]) ²)/n Discrete Random Variable: σ² = _i=1ⁿΣ(x_i-μ)²P(x_i)

Standard Devation/s/σ

For sample: squareroot((_i=1ⁿΣ(x₁-[x bar]) ²)/n-1) For population: squareroot((_i=1ⁿΣ(x₁-[x bar]) ²))/n For discrete random variable: squareroot(σ²) = squareroot(Var(X))

Coefficient of Variation/CV

Used to compare dispersion in data sets with dissimilar units of measurement. 100*(s/[x bar])

Mean Absolute Deviation

The average distance from the center. (_i=1ⁿΣ(x₁-[x bar]) ²)/n

Chebyshev’s Theorem

For any population with mean m and standard deviation s, the percentage of observations that lie within k standard deviations of the mean must be at least 100[1 – 1/k²]. k = 2 then 75%, k = 3 then 88.9%, k = 4 then 93.8%

Empirical Rule

For data from a normal distribution, we expect the interval m ± ks to contain a known percentage of data. k = 1 then 68.26%, k = 2 then 95.44%, k = 3 then 99.73%

Z-score

A measure of how far something is from the mean. - to to the left, + is to the right. z_i = (x_i-μ)/s

Unusual

Data beyond two standard deviations from the mean. Will show up outside the inner fences on a Box Plot.

Estimating Σ

You can estimate the standard deviation as s = R/6 for a normal distribution

Percentiles

Data divided into 100 groups. (Percentile) * n = data point

Quartiles

Data dividede into 4 groups.

Interquartile Range

Indicates variability. Q₃ - Q₁

Box Plot

A visual representation of quartiles 1 through 3, plus outliers and the min/max of the data set. Show variability and shape

Inner Fences

Lower Fence Q₁-1.5(Q₃-Q₁) Upper Fence: Q₃+1.5(Q₃-Q₁)

Outer Fences

Lower Fence: Q₁-3(Q₃-Q₁) Upper Fence: Q₃+3(Q₃-Q₁)

Mid Hinge

The average of the first and third quartiles. (Q₁+Q₃)/2

Covariance

The degree to which the values of X and Y change together. Population: (_i=1ⁿΣ(x_i-[x bar])*(y_i-[x bar]_y)/n Sample: (_i=1ⁿΣ(x_i-[x bar])*(y_i-[x bar]_y)/n-1

Correlation Coefficeint

The covariance divided by the product of the standard deviations. Always between -1 and 1. s_xy/(s_x*s_y) Alternative Sample: (_i=1ⁿΣ(x₁-[x bar])*(y_i-[x bar]_y)/squareroot((_i=1ⁿΣ(x_i-[x bar])²)*squareroot((_i=1ⁿΣ(y_i-[x bar])²)