An individual is an object described by a set of data. A variable is a characteristic we measure one each individual.
There are 2 main types of variables:
- Categorical variable: places individuals into one of several groups or categories
- Quantitative variable: takes numerical values for which it makes sense to find an average
If knowing the value of one variable helps predict the value of the other, it is called an association
A distribution tells us what values the variable takes and how often it takes these values. The 2 distributions we will look at right now are the marginal distribution and the conditional distribution. A marginal distribution (of categorical variables in a two-way table of counts) is a distribution of that variable among all individuals described by the table. Meanwhile, a conditional distribution describes the value of a variable among individuals who have a specific value of another variable. There is a separate conditional distribution for each value of the other variable.
When you are asked to describe a distribution, use the acronym SOCS to structure your response.
- Shape: How many peaks are there on the dotplot? Is there a cluster of dots or a large gap with no dots? What is the most common value (mode)? Look for rough symmetry and skewness.
- Outlier: Are there dots that stand out from the overall pattern of the distribution?
- Center: What is the “midpoint” of the dots? What does this tell us?
- Spread: What is the smallest value? The largest? How far apart are they?
What to look for in shape
A roughly symmetrical distribution has both sides of the graph approximately mirror images to each other. A distribution is skewed to the right if the right tail of the graph is much longer than the left side. It is skewed to the left if the left tail of the graph is much longer than the right side. Remember: The direction of the skew follows the tail.
A unimodal distribution has one peak. A bimodal distribution has two peaks. A multimodal distribution has multiple peaks.
What to look for in center
Generally, the most common measure of center is the ordinary arithmetic average, or mean.
Mean: to find the mean x̄ of a set of observations, add their values and divide by the number of observations. If the n observations are x1, x2,…, xn, their mean is:
x̄ = sum of observations/n
Important note: x̄ refers to a sample mean. μ refers to the population mean. It will be important to keep track of all the symbols used in stats.
Because the mean cannot resist the influence of extreme observations, we say it is not a resistant measure of center. The mean is often called the balance point of a distribution. For example, if you imagine the graph to be a solid object, the place where it balances would likely be the mean.
Median: the midpoint of a distribution, the number such that about half the observations are smaller and about half are larger.
Unlike the mean, the median is a resistant measure of center. No matter how much farther the highest or lowest number is from the rest of the data, the median doesn’t change.
The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median.
What to look for in spread
The IQR, or interquartile range, measures the middle half of a distribution.
IQR = Q3 - Q1 where:
Q1: median of the observations that are to the left of the median in the ordered list. Q3: median of the observations that are to the right of the median in the ordered list.
The IQR is a resistant measure of spread, as it isn’t influenced by a few extreme observations.
What to look for in outliers
Use the 1.5xIQR rule to see if a data point is an outlier. The 1.5xIQR rule states that if a data point is 1.5xIQR higher than Q3 or 1.5xIQR lower than Q1, then that point is an outlier.