places an individual into one of several groups or categories

Quantitative Variable

takes numerical values for which it makes sense to find an average

Distribution

tells us what values a variable takes and how often it takes those values

two-way table

describes two categorical variables, organizing counts according to a row variable and a column variable

marginal distribution

the distribution of values of that variable among all individuals described by the table (for a two-way table)

How to examine a marginal distribution

1) Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals.

2) Make a graph to display the marginal distribution.

conditional distribution

describes the values of that variable among individuals who have a specific value of another variable

How to examine or compare conditional distributions

1) Select the row(s) or column(s) of interest.

2) Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s).

3) Make a graph to display the conditional distribution.
– Use a side-by-side bar graph or segmented bar graph to compare distributions.

How to make a dotplot

1) Draw a horizontal axis (a number line) and label it with the variable name.

2) Scale the axis from the minimum to the maximum value.

3) Mark a dot above the location on the horizontal axis corresponding to each data value.

How to Examine the Distribution of a Quantitative Variable

1) In any graph, look for the overall pattern and for striking departures from that pattern.

2) Describe the overall pattern of a distribution by its:
Shape
Center
Spread

3) Note individual values that fall outside the overall pattern. These departures are called outliers.

symmetric

the right and left sides of the graph are approximately mirror images of each other

skewed to the right (right-skewed)

the right side of the graph (containing the half of the observations with larger values) is much longer than the left side

skewed to the left (left-skewed)

the left side of the graph is much longer than the right side

How to make a stemplot

1) Separate each observation into a stem (all but the final digit) and a leaf (the final digit).
2) Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column.
3) Write each leaf in the row to the right of its stem.
4) Arrange the leaves in increasing order out from the stem.
5) Provide a key that explains in context what the stems and leaves represent.

The most common graph of the distribution of one quantitative variable is a…

histogram

How to make a histogram

1) Divide the range of data into classes of equal width.
2) Find the count (frequency) or percent (relative frequency) of individuals in each class.
3) Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals.

median

the midpoint of a distribution

IQR

Q3 – Q1

To calculate the quartiles:

1) Arrange the observations in increasing order and locate the median.
2) The first quartile Q1 is the median of the observations located to the left of the median in the ordered list.
3) The third quartile Q3 is the median of the observations located to the right of the median in the ordered list.

Rule for Outliers

Call an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile

five-number summary

the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest

How To Make A Boxplot

– A central box is drawn from the first quartile (Q1) to the third quartile (Q3).
– A line in the box marks the median.
– Lines (called whiskers) extend from the box out to the smallest and largest observations that are not outliers.
– Outliers are marked with a special symbol such as an asterisk (*).

Standard deviation (calculation)

square root of the variance

standard deviation (definition)

measures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root.

Variance (calculation)

1) Square each deviation (observation-mean)
2) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)

Variance (defintion)

The average squared distance

Choose a measure of center and spread for a skewed distribution or a distribution with outliers

median and IQR

Choose a measure of center and spread for reasonably symmetric distributions that don’t have outliers

mean and standard deviation

bin/class width

X-scale

Are histograms quantitative or qualitative?

quantitative

Is mean or median more likely to be affected by extremes?

Mean

Is mean or median more resistant to change?

median

Modified boxplot

excludes outliers from whiskers

Population

the entire group of individuals about which we want information about

Census

collects data from every individual in the population

Sample

a subset of the population in which we actually collect data

convenience sample

choosing individuals who are easiest to reach

Bias

A particular preference or point of view that is personal, rather than scientific.

voluntary response sample

people who choose themselves by responding to a general invitation

random sample

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