AP STATS HELL YEAH.

Individual (in statistics)

An object or person described by a data set.

Variable

A characteristic measured or recorded for individuals.

Categorical variable

A variable that places individuals into groups or categories.

Quantitative variable

A numerical variable for which arithmetic operations make sense.

Discrete quantitative variable

A numerical variable with countable possible values.

Continuous quantitative variable

A numerical variable that can take any value within a given interval.

Distribution

The values a variable takes and how often they occur.

Frequency

The number of times a specific value or category occurs in a data set.

Relative frequency

Frequency divided by the total number of observations.

Cumulative relative frequency

The proportion of observations at or below a given value.

Type of data displayed in a bar graph

Categorical data.

Type of data displayed in a pie chart

Categorical data shown as parts of a whole.

Total sum of all slices in a pie chart

$$100\%$$

Type of data displayed in a histogram

Quantitative data grouped into continuous intervals.

Do bars touch in a histogram?

Yes, because the intervals represent a continuous numerical scale.

Do bars touch in a bar graph?

No, because the categories are separate and distinct.

Key difference between a histogram and a bar graph

Histograms display quantitative data across intervals, while bar graphs display categorical data.

Bin (in a histogram)

An interval used to group quantitative observations.

Effect of changing histogram bin width

It can change the apparent shape and display of the distribution.

Dotplot

A display showing each individual observation as a dot above its corresponding value on a number line.

Main advantage of a dotplot

Individual observations and unusual values remain clearly visible.

Primary feature preserved by a stem-and-leaf plot

The original quantitative data values.

In the stemplot key $4 \mid 7 = 47$, what is the stem?

$$4$$

In the stemplot key $4 \mid 7 = 47$, what is the leaf?

$$7$$

Five-number summary

A summary consisting of Minimum, $Q_1$, Median, $Q_3$, and Maximum.

Can the exact mean be determined from a boxplot?

Usually no, because individual data values are not preserved.

CUSS acronym (describing quantitative distributions)

Center, Unusual features, Shape, Spread.

When to use CUSS

When asked to describe the distribution of a quantitative variable.

Center (in CUSS)

A typical or middle value, such as the mean or median.

Unusual features (in CUSS)

Outliers, gaps, clusters, or other notable patterns.

Shape (in CUSS)

The overall form of a distribution (e.g., symmetric, skewed, unimodal, bimodal, uniform).

Spread (in CUSS)

The variability of the data, measured by range, $\text{IQR}$, or standard deviation.

Symmetric distribution

A distribution whose two sides are approximately mirror images of each other.

Relationship between mean and median in a symmetric distribution

They are approximately equal ($\text{mean} \approx \text{median}$).

Right-skewed distribution

A distribution with a long tail extending to the right.

Relationship between mean and median in a right-skewed distribution

The mean is usually greater than the median ($\text{mean} > \text{median}$).

Left-skewed distribution

A distribution with a long tail extending to the left.

Relationship between mean and median in a left-skewed distribution

The mean is usually less than the median ($\text{mean} < \text{median}$).

Direction the mean is pulled in skewed distributions

Toward the long tail.

Unimodal distribution

A distribution with one main peak.

Bimodal distribution

A distribution with two main peaks.

Multimodal distribution

A distribution with multiple main peaks.

Uniform distribution

A distribution where values occur with approximately equal frequency across their range.

Cluster

A group of observations concentrated close together.

Gap

An interval containing few or no observations between data points.

Outlier

An observation located unusually far from the rest of the data.

Mean

The sum of all observations divided by the total number of observations.

Median

The middle observation when data values are arranged in order.

Finding the median with an even number of observations

Calculate the average of the two middle observations.

Mode

The value that occurs most frequently in a data set.

Can a data set have more than one mode?

Yes, a data set can be bimodal or multimodal.

Range formula

$$\text{Range} = \text{Maximum} - \text{Minimum}$$

First quartile ($Q_1$)

The value with about $25\%$ of observations at or below it.

Second quartile ($Q_2$)

The median; the value with about $50\%$ of observations at or below it.

Third quartile ($Q_3$)

The value with about $75\%$ of observations at or below it.

Interquartile Range formula ($\text{IQR}$)

$$\text{IQR} = Q_3 - Q_1$$

Portion of data described by $\text{IQR}$

The spread of the middle $50\%$ of the observations.

Standard deviation

A measure of the typical distance of observations from the mean.

Indication of a larger standard deviation

Greater variability or spread around the mean.

Can standard deviation be negative?

No, it is always greater than or equal to zero.

Condition when standard deviation equals zero

When every observation in the data set has the exact same value.

Variance

The square of the standard deviation ($s^2$ or $\sigma^2$).

Resistant statistic

A summary statistic that is not heavily influenced by extreme values or outliers.

Resistant measures of center and spread

Median and $\text{IQR}$.

Non-resistant measures of center and spread

Mean and standard deviation.

Best measures of center and spread for symmetric data without outliers

Mean and standard deviation.

Best measures of center and spread for skewed data or data with outliers

Median and $\text{IQR}$.

Lower outlier fence formula

$$Q_1 - 1.5 \times \text{IQR}$$

Upper outlier fence formula

$$Q_3 + 1.5 \times \text{IQR}$$

Condition for a low potential outlier

A value that falls below $Q_1 - 1.5 \times \text{IQR}$.

Condition for a high potential outlier

A value that exceeds $Q_3 + 1.5 \times \text{IQR}$.

Are outlier fences required to be actual data values?

No, they are calculated cutoff threshold values.

If $Q_1 = 10$ and $Q_3 = 30$, what is $\text{IQR}$?

$20$ (since $30 - 10 = 20$)

If $\text{IQR} = 20$, what is $1.5 \times \text{IQR}$?

$$30$$

If $Q_1 = 10$ and $\text{IQR} = 20$, what is the lower outlier fence?

$-20$ (calculated as $10 - 30 = -20$)

If $Q_3 = 30$ and $\text{IQR} = 20$, what is the upper outlier fence?

$60$ (calculated as $30 + 30 = 60$)

z-score

A measure of how many standard deviations an observation lies from the mean.

Meaning of a positive z-score

The observation is above the mean.

Meaning of a negative z-score

The observation is below the mean.

Meaning of $z = 0$

The observation is equal to the mean.

Meaning of $z = 2$

The observation is $2$ standard deviations above the mean.

Meaning of $z = -1.5$

The observation is $1.5$ standard deviations below the mean.

Usefulness of z-scores in data comparison

They allow comparison of relative positions across different distributions or scales.

If $\text{mean} = 50$, $\text{SD} = 10$, and $x = 70$, what is $z$?

$2$ (calculated as $\frac{70 - 50}{10} = 2$)

If $\text{mean} = 100$, $\text{SD} = 20$, and $x = 80$, what is $z$?

$-1$ (calculated as $\frac{80 - 100}{20} = -1$)

80th percentile

The value such that about $80\%$ of observations fall at or below it.

Does scoring at the 80th percentile mean getting 80% correct?

No, it describes relative position within a group, not raw percent correct.

Density curve

A mathematical model describing the overall pattern of a continuous distribution.

Total area under a density curve

$1$ (or $100\%$)

Effect of the median on the area under a density curve

It divides the total area under the curve into two equal halves of $0.5$ each.

Normal distribution

A symmetric, unimodal, bell-shaped continuous probability distribution.

Parameter determining the center of a Normal distribution

The mean ($\mu$).

Parameter determining the spread of a Normal distribution

The standard deviation ($\sigma$).

68-95-99.7 Empirical Rule

In a Normal distribution, approximately $68\%$, $95\%$, and $99.7\%$ of observations fall within $1$, $2$, and $3$ standard deviations of the mean.

Percent of data within $1$ standard deviation of the mean in a Normal distribution

Approximately $68\%$

Percent of data within $2$ standard deviations of the mean in a Normal distribution

Approximately $95\%$

Percent of data within $3$ standard deviations of the mean in a Normal distribution

Approximately $99.7\%$

Percent of data lying between the mean and $1\sigma$ above the mean in a Normal distribution

Approximately $34\%$

Percent of data lying between $1\sigma$ and $2\sigma$ above the mean in a Normal distribution

Approximately $13.5\%$

Effect on spread measures when adding a constant to every value

Spread measures (range, $\text{IQR}$, standard deviation) remain unchanged.