Problem Solving and Data Analysis

Mean

The average of a data set found by adding all values and dividing by the number of values.

Median

The middle value of an ordered data set. If there are an even number of values average the two middle numbers.

Mode

The value that appears most often in a data set.

Range

The difference between the greatest and least values in a data set.

Interquartile Range

The difference between the third quartile and the first quartile. It measures the spread of the middle fifty percent of the data.

Quartiles

The values that divide an ordered data set into four equal parts.

First Quartile

The median of the lower half of a data set.

Second Quartile

The median of the entire data set.

Third Quartile

The median of the upper half of a data set.

Five Number Summary

Minimum first quartile median third quartile maximum.

Box Plot

A graph showing the five number summary of a data set.

Outlier

A value much larger or smaller than the rest of the data.

Standard Deviation

Measures how spread out data values are from the mean. A small standard deviation means data are clustered. A large standard deviation means data are more spread out.

Distribution

The overall pattern of a data set.

Symmetric Distribution

The left and right sides are approximately mirror images.

Skewed Right Distribution

Has a long tail extending to the right. Mean is greater than the median.

Skewed Left Distribution

Has a long tail extending to the left. Mean is less than the median.

Cluster

A group of data points close together.

Gap

An interval where no data values occur.

Dot Plot

Displays each data value as a dot on a number line.

Histogram

Displays the frequency of data within intervals.

Frequency

Table or graph showing how often values occur.

Relative Frequency

Frequency divided by the total number of observations.

Two Way Table

Organizes data according to two categorical variables.

Conditional Relative Frequency

The proportion within one row or column of a two way table.

Association

A relationship between two categorical variables.

No Association

Knowing one variable does not help predict the other.

Scatterplot

A graph showing the relationship between two quantitative variables.

Positive Association

As one variable increases the other tends to increase.

Negative Association

As one variable increases the other tends to decrease.

No Association

There is no clear relationship between the variables.

Linear Relationship

The data follow an approximately straight line.

Nonlinear Relationship

The data follow a curved pattern.

Line of Best Fit

The line that best represents the trend in a scatterplot.

Residual

The difference between an observed value and the predicted value.

Residual Formula

Observed value minus predicted value.

Positive Residual

The actual value is above the line of best fit.

Negative Residual

The actual value is below the line of best fit.

Correlation

The strength and direction of a relationship between two variables.

Strong Correlation

Data points lie close to the line of best fit.

Weak Correlation

Data points are widely scattered around the line of best fit.

Correlation Does Not Imply Causation

Just because two variables are related does not mean one causes the other.

Explanatory Variable

The independent variable used to explain changes in another variable.

Response Variable

The dependent variable affected by the explanatory variable.

Prediction

Using a line of best fit to estimate an unknown value.

Interpolation

Predicting within the range of existing data.

Extrapolation

Predicting outside the range of existing data. Less reliable than interpolation.

Random Sample

Every member of the population has an equal chance of being selected.

Population

The entire group being studied.

Sample

A subset of the population.

Sample Size

The number of observations in a sample.

Representative Sample

A sample that accurately reflects the population.

Biased Sample

A sample that does not accurately represent the population.

Voluntary Response Bias

Occurs when participants choose whether to respond.

Selection Bias

Occurs when some members of the population are more likely to be chosen.

Random Assignment

Randomly placing subjects into experimental groups.

Observational Study

Researchers observe subjects without imposing treatments.

Experiment

Researchers apply treatments to determine cause and effect.

Control Group

The group that does not receive the treatment.

Experimental Group

The group receiving the treatment.

Confounding Variable

A variable that influences the results besides the explanatory variable.

Probability

The likelihood that an event will occur.

Probability Formula

Probability equals favorable outcomes divided by total possible outcomes.

Certain Event

Has probability equal to one.

Impossible Event

Has probability equal to zero.

Complement Rule

Probability of not A equals one minus probability of A.

Independent Events

The occurrence of one event does not affect the probability of another.

Dependent Events

The occurrence of one event changes the probability of another.

Probability of Independent Events

Multiply the individual probabilities.

Probability of Either Event

Add probabilities then subtract the overlap.

Mutually Exclusive Events

Events that cannot happen at the same time.

Union of Events

The probability that at least one event occurs.

Intersection of Events

The probability that both events occur.

Replacement

Returning an item before selecting again. Events remain independent.

Without Replacement

Not returning an item. Events become dependent.

Fundamental Counting Principle

Multiply the number of choices for each step.

Permutation

The number of arrangements where order matters.

Combination

The number of selections where order does not matter.

Expected Value

Multiply each outcome by its probability then add the products.

Percent Formula

Percent equals part divided by whole times one hundred.

Percent Increase Formula

Quantity new minus original end quantity divided by original times one hundred.

Percent Decrease Formula

Quantity original minus new end quantity divided by original times one hundred.

Percent Error Formula

Quantity measured minus actual end quantity absolute value divided by actual times one hundred.

Margin of Error

The amount a sample statistic may differ from the true population value.

Units

Always include correct units in the final answer when required.

Ratio

Table comparisons are often expressed as ratios rather than percentages.

Weighted Mean

Multiply each value by its weight add the products then divide by the total weight.

Density

Density equals mass divided by volume.

Population Density

Population divided by land area.

Speed

Speed equals distance divided by time.

Average Speed

Total distance divided by total time not the average of individual speeds.

Fuel Efficiency

Miles divided by gallons or kilometers divided by liters depending on units.

Unit Rate

The amount per one unit.

Conversion Factor

A ratio equal to one used to convert units.

Compound Interest Formula

Final amount equals principal times quantity one plus rate divided by number of compounding periods end quantity raised to the number of compounding periods times time.

Simple Interest Formula

Interest equals principal times rate times time.

Growth Factor

One plus the rate expressed as a decimal.

Decay Factor

One minus the rate expressed as a decimal.

Linear Regression

The equation of the line of best fit used to model data.

SAT Data Strategy

Always read graph labels axes units scales and titles before answering. Many wrong answers result from ignoring units or misreading the graph.