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Credit Sprint's Guide to AP® Statistics, Unit 1: Exploring One-Variable Data and Collecting Data

AP® weight 20–30%

Unit 1 is the foundation for everything else. You learn to describe a distribution (shape, center, variability, unusual features), summarize it with the right statistics, use z-scores and the normal model, and then judge how the data were collected: sampling methods, experiments, bias and what a study can and cannot conclude.

The normal curve and the empirical rule -3σ-2σ-1σmean+1σ+2σ+3σ 68% within 1σ · 95% within 2σ · 99.7% within 3σ

What you need to know

  • Statistics starts with a question. Identify the population, the sample, the variables (categorical or quantitative), and the parameter you want to know versus the statistic you can compute. A good investigative question can be answered with data.
  • Describe a quantitative distribution with four things in context: shape (symmetric, skewed left or right, unimodal or bimodal), center, variability, and unusual features such as gaps or outliers. Use words like 'the typical value is about', not just numbers.
  • Categorical variables use bar graphs, and each bar shows a count or a relative frequency. Quantitative variables use dotplots, histograms and boxplots. A histogram groups values into bins, a dotplot shows every value, and a boxplot shows the five-number summary.
  • Mean and standard deviation are pulled by skew and outliers (not resistant). Median and IQR are resistant. In a right-skewed distribution the mean is greater than the median; in a left-skewed one the mean is less. Use median and IQR when the shape is skewed or has outliers.
  • The standard deviation measures the typical distance of values from the mean. The IQR is Q3 − Q1. A common rule flags a value as an outlier if it is more than 1.5 × IQR below Q1 or above Q3.
  • A z-score tells how many standard deviations a value is from its mean: z = (x − mean) ÷ SD. Adding a constant to every value shifts the center but not the spread; multiplying by a constant multiplies both center and spread.
  • The normal model is symmetric and bell-shaped, described by its mean and standard deviation. The empirical rule gives about 68%, 95% and 99.7% within 1, 2 and 3 standard deviations. For other areas, convert to a z-score and use a table or technology.
  • A census measures everyone. A sample measures part of the population. Random sampling methods are the simple random sample (every group of size n is equally likely), stratified (random samples within groups), cluster (all members of randomly chosen groups) and systematic (every kth member after a random start).
  • Bias is a systematic tendency to miss the truth. Undercoverage leaves out part of the population, voluntary response and convenience samples over-represent people with strong opinions or easy access, nonresponse leaves out people who do not answer, and response bias comes from wording, interviewers or untruthful answers.
  • An observational study records what happens and can only show association. An experiment imposes a treatment on experimental units. Random assignment makes the groups similar so a difference in outcomes can be attributed to the treatment. Good designs use a control or placebo, blinding, replication and sometimes blocking.
  • A confounding variable is related to both the explanatory and response variable and cannot be separated from the treatment's effect. A randomized block design groups similar units first and randomizes within blocks; a matched-pairs design uses pairs (or the same subject twice) and randomizes the order or the member that gets each treatment.
  • Random sampling allows you to generalize from the sample to the population. Random assignment allows you to conclude cause and effect. A study with both can do both; many studies have only one.

Exam traps

Watch forThe mean is not resistant. If a distribution is skewed or has an outlier, the best summary of center is the median and of spread is the IQR.
Watch forAlways describe in context. 'Skewed right' is incomplete; say 'the distribution of commute times is skewed right with a median of 22 minutes.'
Watch forRandom assignment and random sampling are different. Sampling is about who is chosen; assignment is about who gets which treatment.
Watch forA larger sample does not fix bias. A big voluntary response sample is still biased.
Watch forA bar graph and a histogram are different. Bars in a bar graph are for categories and have gaps; histogram bars are bins of a number line.

Key terms

  • Population: The entire group you want to learn about.
  • Sample: The part of the population from which data are actually collected.
  • Parameter: A number that describes a population, such as the true proportion or mean.
  • Statistic: A number computed from a sample, used to estimate a parameter.
  • Categorical variable: A variable that places each individual in a group or category.
  • Quantitative variable: A variable that takes numerical values for which arithmetic makes sense.
  • Distribution: The values a variable takes and how often it takes them.
  • Relative frequency: The count in a category divided by the total, often as a percent.
  • Shape: Whether a distribution is symmetric or skewed, and whether it has one peak or more.
  • Skewed right: A distribution with a long tail toward larger values; the mean is usually greater than the median.
  • Skewed left: A distribution with a long tail toward smaller values; the mean is usually less than the median.
  • Symmetric: A distribution whose left and right sides are roughly mirror images.
  • Unimodal and bimodal: Having one peak or two distinct peaks.
  • Outlier: A value that falls far from the rest of the distribution.
  • Mean: The balance point of the data: the sum of the values divided by n.
  • Median: The middle value of the ordered data, so half the values are below it.

Practice questions

Five of the 128 Unit 1 questions in Credit Sprint. Every choice is explained in the app.

1. A dotplot of the number of siblings for 30 students is strongly skewed to the right. Which pair of statistics best describes center and spread?
  1. Mean and standard deviation
  2. Mode and variance
  3. Median and IQR
  4. Mean and range
Show answer

Median and IQR. For a skewed distribution, the median and IQR are resistant to the long tail, so they describe the typical value and spread better.

2. What does a boxplot show that a histogram does not show directly?
  1. The five-number summary, including the median and quartiles
  2. Every individual data value
  3. The exact shape of every peak in the data
  4. The relationship between two variables
Show answer

The five-number summary, including the median and quartiles. A boxplot displays the minimum, Q1, median, Q3 and maximum, so these values can be read directly.

3. A researcher wants a sample of 200 students from a school of 2,000. She numbers the students 1 to 2,000 and uses a random number generator to pick 200 different numbers. This is a
  1. simple random sample
  2. systematic sample
  3. stratified random sample
  4. cluster sample
Show answer

simple random sample. Every group of 200 students has the same chance of being chosen, which is the definition of an SRS.

4. A poll asks, 'Do you support the new law that protects children from harmful online content?' This question is most likely to produce
  1. response bias from leading wording
  2. nonresponse bias
  3. sampling variability
  4. undercoverage
Show answer

response bias from leading wording. The wording pushes respondents toward supporting the law, which distorts the true level of support.

5. An experiment tests whether a new fertilizer increases tomato yield. The researcher notes that soil quality varies a lot across the field. Which design best controls for this?
  1. Use the new fertilizer on all plots
  2. Let farmers choose which plots to fertilize
  3. Divide the field into blocks of similar soil and randomly assign treatments within each block
  4. Give the new fertilizer to the best soil
Show answer

Divide the field into blocks of similar soil and randomly assign treatments within each block. Blocking on soil quality removes that source of variability so the effect of the fertilizer is easier to see.