# Mathematics Common Core Standards: Statistics and Probability

*Statistics* (which analyzes existing data) and *probability* (which uses existing data to predict future events) are two branches of math needed for Common Core Standards that students can see at work in the world.

Statistics are used in everything from gauging how citizens feel about a certain politician to setting insurance premiums to informing the debate about climate change. Probability is often used to make decisions, such as when to plant a certain crop, whether a business should expand, or whether an individual should run for political office.

## How to interpret categorical and quantitative data

Expectations in this area call on students to gather, analyze, and present two types of data:

**Categorical:**Categorical data is often used to compare and contrast groups; for example, one study shows that the most popular car color is white. Silver and black are tied for second.**Quantitative:**Quantitative data represents measurements, such as length, number of votes, population density, and so forth.

Students display data in various forms, including number lines, graphs, and charts using various measures of the center and methods to determine patterns, repetition, and trends in data. Some common terms that you're likely to encounter are

*mean*(the average)*median*(the middle number when data is organized from least to greatest)*standard deviation*(a description of the distance from the center in a collection of data)*correlation*(when the frequency or occurrence of two things is related)*causation*(when something causes another event to happen)

Here's an example of a typical problem that requires the use of existing data to make predictions regarding future situations: Imagine that a bank is busiest from 4 p.m. to 6 p.m. on weekday evenings. During these hours, the wait time in the drive-through is normally distributed, with a mean of 8 minutes and a standard deviation of 2 minutes.

Using standard deviations, determine a) the percentage of customers who wait 10 minutes or longer, b) the percentage who wait between 4 and 12 minutes, and c) the percentage who wait 2 minutes or less.

Draw a standard bell curve and then do the math:

10 minutes or longer:

Add the percentages in the 10–12, 12–14, and >14 ranges: 13.6 + 2.2 + 0.1 = 15.9 percent

4 to 12 minutes:

Add the percentages in the 4–6, 6–8, 8–10, and 10–12 ranges: 13.6 + 34.1 + 34.1 + 13.6 = 47.7 + 47.7 = 95.4 percent

Less than 2 minutes:

Take the percentage in the <2 range: 0.1 percent

## Make inferences and justify conclusions

Students discover statistics as a way to find out about a population or group without necessarily gathering information from every person in that population. This includes making *inferences* — conclusions based on evidence. When looking at methods for making determinations about populations of events using statistical methods, students discuss whether the methods are reliable — for example, whether the people polled are actually representative of the entire population.

Students also explore uses of *randomization* to improve the accuracy of data. For example, in clinical trials of new medications, participants in the study are almost always chosen randomly for the two groups — the group that receives the medication and the other group that gets the placebo, for example. This approach lessens the chance that some other factor will skew the results.

For example, if one group consisted exclusively of men and the other of women, results could be influenced by the sex of the participants rather than by whether the medication was more effective than the placebo.

With your child, examine a recent poll conducted on a political issue. Discuss all of the components used in gathering data, such as the size of the sample population, the means of gathering data, and the interpretation of the results. Share your opinions on the reliability of the conclusions drawn from the data.

Identify instances when randomizing data collection is appropriate for statistical validity and to remove the potential for bias (in other words, to ensure that the data isn't slanted in any particular direction).

## Conditional probability and probability rules

High-school math includes the study of conditional probability — that is, the likelihood that the outcome of one event will influence the outcome of another event. Students explore techniques for determining whether two events are *independent* (neither event influences the other event) or *conditional* (the probability of one event occurring is influenced by whether the other event occurs).

Students also discover how to use data to predict the likelihood of certain events when multiple options are involved. The probability of compound events (when the same trial is attempted multiple times with the same circumstances) is also addressed in these standards.

Here's a sample problem: You randomly draw two cards from a standard deck of 52 playing cards. What are your odds of drawing two clubs?

A deck of cards has 13 clubs, meaning you have a 13 in 52 or 1 in 4 chance of drawing a club as the first card. For the second draw, only 12 clubs remain out of 51 total cards. This results in a 4 out of 17 chance that you'll pull a club on the second draw.

To determine the probability of this occurring in consecutive draws, multiply the two ratios:

You have a 1 in 17 chance of drawing two clubs consecutively.

## Use probability to make decisions

One of the most compelling reasons to develop a knack for calculating probabilities is because this skill often enables you to make better decisions. Students use probability to assess the likelihood of the number of occurrences or events within a set of data and then use that information to answer questions or draw conclusions based on the results.

Students also make use of probability to determine the outcome of events based on chance and to analyze decision-making in certain scenarios.