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Statistics

The science of collecting, organizing, analyzing, interpreting, and presenting data.

Measure of Central Tendency

A statistic that describes the center or typical value of a distribution.

Mean

The arithmetic average found by adding all values and dividing by the number of values.

Arithmetic Mean

The most commonly used measure of central tendency; calculated as ΣX ÷ N.

Weighted Mean

A mean that assigns different levels of importance (weights) to different values.

Weight

A numerical value assigned to indicate the relative importance of a score.

Median

The middle score in an ordered distribution.

Median (Odd Number of Scores)

The exact middle value when data are arranged by magnitude.

Median (Even Number of Scores)

The average of the two middle values when data are arranged by magnitude.

Mode

The most frequently occurring value in a distribution.

Unimodal Distribution

A distribution with one mode.

Bimodal Distribution

A distribution with two modes.

Multimodal Distribution

A distribution with more than two modes.

Measure

A value that is representative of a set of data.

Representative Value

A value that best describes the center of a distribution.

Distribution

An arrangement of data showing how frequently different values occur.

Frequency Distribution

A table showing values and the number of times they occur.

Normal Distribution

A symmetrical bell-shaped distribution where mean, median, and mode are equal.

Bell-Shaped Curve

The graphical appearance of a normal distribution.

Bilaterally Symmetrical Curve

A curve that can be folded down the center and match on both sides.

Symmetrical Distribution

A distribution with equal shape on both sides of the center.

Peaked Curve

A symmetrical curve that is taller and narrower than a normal curve.

Flat Curve

A symmetrical curve that is shorter and wider than a normal curve.

Skewed Distribution

A distribution that is asymmetrical.

Positive Skewness

A distribution with a tail extending to the right.

Right-Skewed Distribution

Another term for positive skewness.

Negative Skewness

A distribution with a tail extending to the left.

Left-Skewed Distribution

Another term for negative skewness.

Effect of Positive Skewness on Mean

The mean is pulled toward the right tail.

Effect of Negative Skewness on Mean

The mean is pulled toward the left tail.

Effect of Skewness on Mode

The mode shifts toward the peak of the distribution and away from the tail.

Effect of Skewness on Median

The median lies between the mean and mode.

Best Measure for Normal Data

Mean.

Best Measure for Skewed Data

Median.

Best Measure for Nominal Data

Mode.

Outlier

An unusually high or low value compared with the rest of the data.

Effect of Outliers on Mean

Outliers significantly affect the mean.

Effect of Outliers on Median

Outliers have little effect on the median.

Trimmed Mean

A mean calculated after removing extreme values.

J-Shaped Distribution

A distribution where frequencies increase sharply at one end.

Reverse J-Shaped Distribution

A distribution where frequencies decrease sharply at one end.

Dispersion

The spread or variability of data.

Variation

Another term for dispersion.

Variability

The degree to which values differ from one another.

Homogeneous Data

Data with relatively little variation.

Heterogeneous Data

Data with substantial variation.

Measure of Dispersion

A statistic describing the spread of data.

Range

The difference between the highest and lowest values.

Range Formula

Highest value − Lowest value.

Advantage of Range

Easy to calculate.

Disadvantage of Range

Uses only the highest and lowest values.

Variance

The average squared deviation from the mean.

Population Variance

Variance calculated using all members of a population.

Sample Variance

Variance calculated using a sample from a population.

Standard Deviation

The square root of the variance.

Purpose of Standard Deviation

Measures the average distance of values from the mean.

Large Standard Deviation

Indicates greater variability.

Small Standard Deviation

Indicates less variability.

Absolute Variability

Variability measured without reference to the mean.

Relative Variability

Variability measured relative to the mean.

Coefficient of Variation (CV)

The ratio of the standard deviation to the mean expressed as a percentage.

Coefficient of Variation Formula

(Standard Deviation ÷ Mean) × 100.

Purpose of Coefficient of Variation

Compare variability between data sets with different units.

Grouped Data

Data organized into classes or intervals.

Ungrouped Data

Data listed individually without intervals.

Grouped Mean

Mean estimated using class midpoints and frequencies.

Class Midpoint

The value halfway between the class limits.

Midpoint Formula

(Lower Limit + Upper Limit) ÷ 2.

Grouped Variance

Variance calculated using class midpoints and frequencies.

Grouped Standard Deviation

Standard deviation calculated from grouped data.

Empirical Rule

The rule describing the relationship between standard deviations and the area under a normal curve.

68-95-99 Rule

Another name for the empirical rule.

One Standard Deviation Rule

Approximately 68% of values fall within ±1 SD of the mean.

Two Standard Deviations Rule

Approximately 95% of values fall within ±2 SD of the mean.

Three Standard Deviations Rule

Approximately 99% of values fall within ±3 SD of the mean.

Normal Curve

A distribution defined by its mean and standard deviation.

Area Under the Curve

The proportion of observations represented in a distribution.

Percentile

A value below which a certain percentage of observations fall.

99th Percentile

A value greater than approximately 99% of observations.

Chebyshev’s Theorem

A theorem that applies to any distribution shape and estimates the minimum proportion of data within k standard deviations.

Chebyshev’s Theorem Requirement

k must be greater than 1.

Chebyshev’s Theorem Purpose

Estimate the minimum percentage of values within a specified number of standard deviations.

Raw Data

Data recorded in the order collected before organization.

Table

A systematic arrangement of data in rows and columns.

Table Number

An identifier used to reference a table.

Table Title

A description explaining what, where, and when the data were collected.

Stub Heading

The heading of the first column in a table.

Column Heading

A label identifying data within a column.

Data Cell

The location in a table where data values are recorded.

Footnote

Additional information explaining table content.

Units of Measure

Labels indicating how data are measured.

Frequency

The number of times a value occurs.

Qualitative Data

Non-numerical data classified into categories.

Quantitative Data

Numerical data that can be measured.

Qualitative Frequency Distribution

A table showing categories and their frequencies.

Relative Frequency

Frequency divided by the total number of observations.

Relative Frequency Formula

Frequency ÷ Total Frequency.

Frequency Percentage

Relative frequency multiplied by 100.

Frequency Percentage Formula

Relative Frequency × 100.

Total Relative Frequency

Always equals 1.00.