Interpret statistics in context, judge whether a measure or claim is appropriate, recognize misleading displays and biased samples, and reason about how outliers affect summaries.

What this topic is asking

The Statistics and Probability category expects you to interpret statistics critically (the S-IC and S-ID standards). On the Grade 10 MCAS this means judging whether a measure or a claim fits the data, spotting misleading displays and biased samples, and reasoning about how outliers distort summaries. These are reasoning and short-answer items where a clear, correct explanation earns the credit.

Choosing the appropriate measure

A summary should represent the data honestly. The main decision is between mean and median:

• Use the mean for roughly symmetric data with no extreme values.
• Use the median when the data is skewed or has outliers, because the mean is dragged toward the extremes while the median stays in the middle of the bulk.

For incomes, house prices, or any data with a few very large values, the median is usually the fairer "typical" value. The MCAS often asks you to justify the choice, so naming the outlier or the skew and explaining its effect on the mean is the substance of a full answer.

How outliers affect summaries

An outlier is a value far from the rest. Its effect depends on the statistic:

• Mean: strongly affected, because the mean adds every value, so one extreme value shifts it noticeably.
• Median: barely affected, because it depends on the position of the middle value, not its size.
• Range: strongly affected, since it uses the extremes.
• IQR: barely affected, since it uses the middle 50%.

So adding a single large outlier raises the mean and the range a lot but moves the median and IQR little, which is the reasoning the MCAS tests.

Misleading displays

Graphs can distort even when the numbers are correct. The MCAS asks you to spot tricks:

• A truncated vertical axis (one that does not start at zero) makes small differences look huge. A bar chart of values 102 and 104 with an axis from 100 to 105 exaggerates a tiny gap.
• Uneven or inconsistent scales misrepresent rates of change.
• A misleading pictograph, where larger icons imply more than the count warrants, overstates differences.

Reading the axis labels and scale before trusting the visual impression is the defense.

Biased samples

A conclusion about a population is only as good as the sample behind it. A sample is biased when it systematically over- or under-represents part of the population:

• Convenience or location bias: surveying only people at one place (a park, a store) misses everyone else.
• Voluntary response bias: people who choose to respond often hold stronger opinions.
• Self-interest: a survey run by a party with a stake can be skewed by question wording.

A random sample, where every member has an equal chance of selection, is what supports a valid generalization. The MCAS tests whether you can identify why a given sampling method is biased.

Try this

Q1. Data has one very large outlier. Which is more representative, mean or median?

• Cue. The median.

Q2. A poll surveys only a website's own users about that website. What bias is this?

• Cue. Voluntary response or sampling bias (not representative).