SAT Math 9: One-Variable Data — Measures of Center and Spread

Skewness

Skewness describes the lack of symmetry in a distribution and identifies the direction in which a distribution has a longer tail.

Symmetric distribution

A distribution is symmetric when a vertical line can divide its graph into two mirror-image halves. In a perfectly symmetric distribution, the mean and median are equal or approximately equal.

Unimodal distribution

A distribution with one mode, meaning one value or interval occurs with the greatest frequency.

Bimodal distribution

A distribution with two modes or two prominent high points. In a symmetric bimodal distribution, the mean and median may lie between the modes rather than equal either mode.

What relationship among mean, median, and mode is expected for a symmetric, unimodal distribution?

The mean, median, and mode are often equal or very close, with all three near the center and peak of the distribution.

How does a left-skewed distribution usually affect the mean and median?

A left-skewed distribution has a longer tail toward smaller values, so unusually low values pull the mean downward. Typically, mean < median, although this relationship is not guaranteed for every data set.

How does a right-skewed distribution usually affect the mean and median?

A right-skewed distribution has a longer tail toward larger values, so unusually high values pull the mean upward. Typically, mean > median, although this relationship is not guaranteed for every data set.

Which measure of center is generally most affected by outliers: mean, median, or mode?

The mean is generally affected the most because every value contributes to its calculation. The median and mode are usually more resistant to extreme values.

In a skewed distribution, which measure of center is often closest to the high point of the distribution?

The median is often closer to the distribution's high point or mode, while the mean is pulled farther toward the long tail.

How can the shape of a distribution help identify whether the mean or median is the better center?

For skewed data or data with outliers, the median is usually a more representative measure of center because it is resistant to extreme values. For roughly symmetric data without strong outliers, the mean is often appropriate.

How does a histogram complement a numerical measure such as standard deviation?

A histogram displays the distribution's shape, including skewness, peaks, gaps, and possible outliers. This context is important because standard deviation alone may be misleading for skewed data.

Mean

The mean is the arithmetic average of a data set: add all values and divide by the number of values. For a population, $\mu=\frac{\sum x}{N}$; for a sample, $\bar{x}=\frac{\sum x}{n}$.

Median

The median is the middle value when data are ordered. If there are two middle values, the median is their average.

Mode

The mode is the value that occurs most often. A data set can have one mode, more than one mode, or no mode.

Range

The range measures the total spread of a data set: $\text{range}=\text{maximum}-\text{minimum}$.

What happens to the mean, median, and range when an extreme high outlier is added?

The mean usually increases substantially, the range increases, and the median may change little or not at all.

What happens to the mean, median, and range when the same constant is added to every data value?

The mean and median each increase by that constant, and the range does not change.

Relative frequency

The relative frequency of a category or value is its frequency divided by the total number of observations: $\text{relative frequency}=\frac{f}{n}$.

Cumulative relative frequency

Cumulative relative frequency is the running total of relative frequencies up to a given value or interval. It represents the proportion of observations at or below that point.

First quartile and third quartile

$Q_1$ is the median of the lower half of an ordered data set, and $Q_3$ is the median of the upper half. They mark approximately the 25th and 75th percentiles.

Interquartile range (IQR)

The interquartile range measures the spread of the middle 50% of data: $IQR=Q_3-Q_1$. It is resistant to outliers.

Five-number summary

A five-number summary consists of the minimum, first quartile $Q_1$, median, third quartile $Q_3$, and maximum.

What information does a box plot display?

A box plot displays the five-number summary: the minimum and maximum, the first and third quartiles, and the median. The box represents the middle 50% of the data, whose width corresponds to the IQR.

Why are box plots and quartiles especially useful for skewed data?

They show the median, quartiles, and extremes without being strongly distorted by outliers. This makes them useful when the two sides of a skewed distribution have different spreads.

What does a longer whisker on one side of a box plot suggest?

It suggests greater spread on that side of the distribution and often indicates skewness in that direction. For example, a longer left whisker is consistent with left skew.

How can a box plot indicate a potential outlier?

Using the common 1.5-IQR rule, values below $Q_1-1.5(IQR)$ or above $Q_3+1.5(IQR)$ are potential outliers and are often plotted separately from the whiskers.

Histogram

A histogram groups numerical data into intervals, or bins, and uses adjacent bars to show the frequency or relative frequency in each interval.

How is a histogram different from a bar graph?

A histogram displays quantitative data grouped into numerical intervals, so its bars touch. A bar graph usually displays categorical data, so its bars are separated.

What features of a distribution can a histogram reveal?

A histogram can reveal the center, spread, shape, skewness, peaks, gaps, clusters, and possible outliers of a data set.

Deviation from the mean

For a population, a value's deviation is $x-\mu$; for a sample, it is $x-\bar{x}$. It describes how far and in which direction the value is from the mean.

Why are deviations squared when calculating variance?

Positive and negative deviations otherwise cancel, producing a sum of zero. Squaring makes every contribution nonnegative while giving greater weight to larger deviations.

Variance

Variance is the average of the squared deviations from the mean. Population variance is denoted by $\sigma^2$ and sample variance by $s^2$; variance has squared units.

Population variance formula

For a population of size $N$, $\sigma^2=\frac{\sum (x-\mu)^2}{N}$.

Sample variance formula

For a sample of size $n$, $s^2=\frac{\sum (x-\bar{x})^2}{n-1}$.

How are variance and standard deviation related?

Standard deviation is the square root of variance: $\sigma=\sqrt{\sigma^2}$ for a population and $s=\sqrt{s^2}$ for a sample.

Standard deviation

A measure of the typical distance of data values from their mean. It is always nonnegative and has the same units as the original data.

How does a data set's standard deviation change when its values are more widely spread around the mean?

The standard deviation increases as the data become more dispersed. A smaller standard deviation indicates that values are concentrated closer to the mean.

What does a standard deviation of zero imply about a data set?

Every data value is identical, so there is no variation or spread.

Population standard deviation formula

For a population of size $N$, $\sigma=\sqrt{\frac{\sum (x-\mu)^2}{N}}$.

Sample standard deviation formula

For a sample of size $n$, $s=\sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}$.

When should $\sigma$ versus $s$ be used for standard deviation?

Use $\sigma$ when the data represent the entire population and divide by $N$. Use $s$ when the data are a sample and divide by $n-1$.

Why does the sample standard deviation use $n-1$ instead of $n$ in the denominator?

Dividing by $n-1$ corrects for the tendency of a sample to underestimate the population variance. This produces a better estimate of the population variance.

What is the effect of an outlier on standard deviation?

An outlier can make the standard deviation substantially larger because its squared deviation from the mean is large. Standard deviation is therefore not resistant to outliers.

Why can standard deviation be less informative for a skewed distribution?

Standard deviation summarizes spread around the mean, but skewed distributions have different spreads on their two sides. A box plot and the five-number summary may describe such data more effectively.

What does it mean for a value to be one standard deviation above or below the mean?

A value one standard deviation above the mean equals the mean plus the standard deviation; a value one standard deviation below equals the mean minus the standard deviation.