Statistics 201 Exam 1 Study Guide (Chapter 1-6)

What are Statistics (noun)?

numbers, values, data, calculations made from data

What are Statistics (verb)?

a way of reasoning, a collection of tools and methods, designed to help us understand the world

What are Data?

Data can be numbers, characters, images, or other labels

Data are useless without their context

Data in a context

When the data are organized, given labels, and put into a context, the data gain meaning

Population

Entire group a researcher wants to study (impractical or impossible)

Sample

Used to make inferences about the population

Representative

The sample should accurately mirror key characteristics of a larger group (population)

Randomness

A tool to help select a representative sample from a population

The “W’s” to provide context of the data

-Who (essential)

-What (essential)

-When

-Where

-Why (if possible)

-How

Categorical (or qualitative) variable

Possible values are given by short descriptors

-Gender (male/female), eye color (blue/brown), zip codes

-Can encode as numbers (e.g. female=1, male=2)

Special designations of Categorical Variables

Ordinal data - categorical data with an inherent order (e.g., ranks in military, freshman/sophomore/junior/senior)

Identifier data - categorical data with a unique value for each observation (Social Security Number, FedEx Tracking Number, ISBN)

Quantitative Variable

Possible values must be numbers, usually with units

-Income ($), height (inches), weight (pounds)

-Ratio of forearm length to hand length (unitless)

-Common business analytics quantitative variables are profit, net increase, tax rate, days in inventory, price

Simulation

Mimics reality by using random numbers to represent outcomes of real events

Types of Univariate Categorical Displays

-Frequency and Relative Frequency Table

-Bar Chart

-Pareto Chart

-Pie Chart

-Ring Chart

Frequency Tables

-The name of each category, and

-the number of observations in each category

Relative Frequency Tables

Similar to frequency tables, but gives the percentages instead of counts for each category

Bar Chart

Displays the frequency or relative frequency of each category

-All bars must have the same width

-Good for a general audience

Pie Chart

Presents each category as a slice of a circle so that each slice has a size that is proportional to the whole in each category

Ring (or donut) Chart

Presents each category as a partition of a ring that is proportional in area to the value of each category

Pareto Chart

A bar chart with the bars put in order from the most frequently occurring category on the left to the least frequently occurring category on the right

Types of Univariate Quantitative Displays

-Histograms

-Stem and Leaf Plots

-Dotplots

Histrogram

Plots quantitative data by using bins to sort the data points

-When all data are sorted, the histogram visualized the distribution of the quantitative data

Stem-and-Leaf Displays

Show the distribution of a quantitative variable, like histograms do, while preserving the individual values

-The shape of a stem-and-leaf plot is exactly the same as a histogram, except it is in a vertical format

Dotplots

Displays dots to describe the shape of the distribution

When describing a distribution, what three things always need to be told about?

-Shape

-Center

-Spread

Shape

1. Does the histogram have a single, central hump, or several separated humps? (one peak = unimodal, two peaks = bimodal, more than two = multimodal) ALSO: Uniform histogram has close to the same frequency and is nearly flat

2. Is the histogram symmetric? (can you fold the histogram along a vertical line through the middle and have the edges match closely) ALSO: Skewed left or Skewed right based on tails)

3. Do any unusual features stick out? (stragglers or outliers that stand off away, gaps in the distribution)

Center

-The median is the value with exactly half the data values below it and half above it (outliers have no impact on the median because it is based on the position)

-When we have symmetric data, there is an alternative to the median: the average or mean (mean feels like the center because it is the point where the histogram balances)

Mean or Median?

-The median is resistant to values that are extraordinarily large or small, but the mean is not

Which measure of center should you use?

-If the histogram is symmetric and there are no outliers, use the mean

-If the histogram is skewed or with outliers, use the median

Spread

-Are the data near the center or are they spread out? Is the highest value much higher than the lowest value?
-Ways to find: Range, Percentiles, Quartiles, IQR, Standard Deviation

Range

The difference between the maximum and minimum values of the data

-A disadvantage of the range is that a single extreme value can make it very large and thus, not representative of the data overall

Percentiles

Divide the data into percentages with reference to a particular direction

-The nth percentile is the data value such that n percent of the data lies below that value

Quartiles

Divide the data into four equal sections

-One quarter (25%) of the data lies below the lower quartile, Q1

-Three quarters (75%) of the data lies below the upper quartile, Q3

Interquartile Range (IQR)

Let’s us ignore extreme data values and concentrate on the middle of the data

-The difference between the quartiles is the interquartile range (IQR), so

IQR = Q3 - Q1

Thus, the IQR represents the middle 50% of the data

Standard Deviation

A more powerful measure of spread than the IQR which takes into account how far each data value is from the mean

-A deviation is the distance that a data value is from the mean

-The variance is found by summing the squared deviations and averaging them

Types of Bivariate Categorical Data Displays

-Contingency Tables

-Side by Side Pie Charts

-Segmented Bar Charts

-Mosaic Plot

Contingency Table

Allows us to look at two categorical variables together

-Each cell of the table gives the count for a combination of values of the two variables

-Each frequency distribution is called a marginal distribution of its respective variable

Conditional Distributions

Shows the distribution of one variable for just the individuals who satisfy some condition on another variable

Independence

-When the distribution of one variable is the same for all categories of another, we define these variables as independent

-If the distribution of one variable is different from the categories of another, we define these variables as having an association

Segmented Bar Chart

Displays the same information as side by side pie charts, but in the form of bars instead of circles

Mosaic Plot

Has bar widths proportional to the frequencies in the category on the horizontal axis

5-number Summary

Provides a useful overview about a quantitative variable

-Maximum value

-Q3: Third quartile (75th Percentile)

-Median

-Q1: First quartile (25th Percentile)

-Minimum value

Boxplots

A graphical visualization of a quantitative variable

-They are constructed from a 5-number summary

What is a Z-Score

z tells us how many standard deviations the value y is away from the mean

If you convert an entire data set into z-scores:

This is known as standardizing the data

-The z-scores are unitless numbers

-This shifts the mean to 0, and rescales the standard deviation to 1

Shifting and rescaling a data set does not change the shape of the distribution

-This applies to converting data into z-scores, or converting data from one unit of measure to another

Normal Model

The 96-95-99.7 Rule

Normal models are appropriate for distributions whose shapes are unimodal and symmetric

When is a z-score Big?

Remember when we standardized, the mean shifts to 0 and standard deviation shift to 1:
-The N(0,1) model is called the Standard Normal Model (or the Standard Normal Distribution)

Things to Look For in Scatterplots

-Direction (positive, negative, or neither?)

-Form (Approximately a straight line, or something else?)

-Strength (Strong or Weak relationship?)

-Unusual features (Outliers, Clusters or Subgroups)

Roles for Variables

Response Variable (y): The variable of interest. It is what we want to predict.

Explanatory or Predictor Variable (x): The variable that we use to provide information or a prediction of the response variable.

Correlation Conditions

The correlation coefficient {r}

Before you use correlation, you must check several conditions:

-Quantitative Variables Condition

-Straight Enough Condition

-No Outliers Condition