Exploring One-Variable Data and Collecting Data

Statistics lets you answer an investigative question about a large group (the population) by collecting data from a smaller group (the sample), since getting data from every member is usually impossible or too costly. A good question has a clear purpose, doesn't change once you see the results, and can be answered with data you're able to collect.

An observational unit is the item or person you collect data from, and a variable is a characteristic that can differ from unit to unit: categorical (group labels) or quantitative (counted or measured numbers, which can be discrete or continuous). A parameter summarizes a whole population, while a statistic summarizes a sample and is often used to estimate the parameter.

A frequency table counts how many units fall in each category of a categorical variable, and a relative frequency table shows each category's proportion of the total. Proportions, percentages and relative frequencies carry the same information, and you use them to back up claims about the variable.

Bar charts and pie charts show the counts or proportions in each category of one categorical variable: a bar's height or a slice's share of the circle matches that category's share of the data. You can use tables and graphs like these to compare two or more groups on the same categorical variable.

Dotplots, stem-and-leaf plots (stemplots) and histograms show the distribution of a quantitative variable while keeping the values in order from smallest to largest. A histogram groups values into intervals called bins, and changing the bin width can change how the graph looks.

To describe a quantitative distribution, discuss its shape, center, variability (spread) and any unusual features such as outliers, gaps or clusters, always in context. Common shapes are skewed right (a longer right tail), skewed left, roughly symmetric, unimodal, bimodal and approximately uniform.

The mean (x̄) and median measure center; the range, the interquartile range (IQR = Q3 − Q1) and the standard deviation (s) measure variability; and percentiles and quartiles describe a value's position. One common rule calls a value an outlier if it is more than 1.5 × IQR below Q1 or above Q3 (another flags values more than 2 standard deviations from the mean). Outliers barely move the median and IQR, so those are called resistant, while the mean, range and standard deviation are not.

A boxplot graphs the five-number summary (minimum, Q1, median, Q3, maximum), with the box covering the middle 50% of the data; when there are outliers, the whiskers stop at the most extreme values that aren't outliers, and the outliers get their own marks. Shape links the mean and median: in a right-skewed distribution the mean is usually greater than the median, and in a left-skewed one it is usually less.

To compare distributions, compare their shapes, centers, variability and unusual features using comparison words like "greater than," not just separate lists of numbers. A z-score, z = (x − μ)/σ, tells how many standard deviations a value sits above or below the mean, so you can compare values that come from different distributions.

The investigative question gets more detailed here: it should point to the variables to collect, the analysis to run (such as a test or a confidence interval) and the kind of conclusion you can make. You'll also tell apart a census (data from every member), an observational study (no treatments imposed, so a confounding variable could explain the results) and an experiment (treatments assigned to experimental units), and you can only generalize to the whole population when the sample was chosen at random.

Random sampling uses a chance process, like a random number generator, to choose the sample. The main methods are the simple random sample (every possible sample of size n is equally likely), the stratified random sample (an SRS from each group of similar units), the cluster sample (randomly choose whole groups and use everyone in them) and the systematic sample (a random start, then every kth unit).

Bias means the way a sample is chosen pushes a statistic in one direction, so it keeps coming out too high or too low compared with the parameter you're trying to estimate. Watch for voluntary response bias, undercoverage, nonresponse and response bias (such as leading questions), and remember that convenience and volunteer samples aren't random, so they invite bias.

A well-designed experiment compares at least two treatments, assigns them at random, uses replication (more than one unit per treatment) and keeps other sources of variation under control. You'll learn completely randomized, randomized block and matched pairs designs, along with control groups, placebos and blinding, and why random assignment is what allows cause-and-effect conclusions.