Mean, median, mode and spread

01 / UNDERSTAND & APPLY

Identify observations, frequencies and units

A data set is a collection of observations of a specified variable. A frequency tells how often a value occurs; repeated values count repeatedly when finding totals and centers. Record the observation count before dividing. A list of distinct values without their frequencies may not determine the mean.

Units matter: a mean of times is a time; a standard deviation of times is also a time. Variance, the square of standard deviation, uses squared units. A summary does not retain all information about individual observations.

02 / UNDERSTAND & APPLY

The mean balances the total across observations

For n observations, mean=sum/n, so sum=n·mean. This equivalent form makes missing-value and combined-group problems easier. The mean need not be one of the observed values, nor an integer even for integer data.

If a value x is added to a set of n values with mean m, the new mean is (nm+x)/(n+1). Adding a value equal to the mean leaves the mean unchanged; adding one above it raises it, and adding one below lowers it.

03 / UNDERSTAND & APPLY

Weight each value by its number of observations

A weighted mean is Σ(weight×value)/Σweights. For group means, the weights are group sizes. The mean of group means is valid only if group sizes are equal, or a special coincidence makes it so. Positive weights force the result to lie between the smallest and largest group means.

Frequency tables use exactly the same method: value×frequency contributes to the total, and frequencies add to the number of observations. Do not divide by the number of table rows.

04 / UNDERSTAND & APPLY

Position and frequency answer different questions

Sort data before finding the median. With odd n, it is the middle value at position (n+1)/2. With even n, it is the average of values at positions n/2 and n/2+1. The median splits the ordered positions; tied values can appear on both sides.

The mode is a most frequent value. A set can have multiple modes, and depending on the stated convention can have no mode when nothing is more frequent. The mode describes frequency, not numerical size. Mean, median and mode can coincide, but need not.

05 / UNDERSTAND & APPLY

The range uses only two observations

Range=maximum−minimum and is nonnegative. It uses only the endpoints, so two sets can share a range while clustering very differently inside it. An extreme observation can greatly increase range while leaving median unchanged.

Spread and center are separate. Translating every value by the same amount moves the center but leaves pairwise distances unchanged. A set with a larger mean need not have a larger range or standard deviation.

06 / UNDERSTAND & APPLY

Standard deviation measures typical deviation from the mean

For a full finite distribution treated as a population, variance is the mean of squared deviations from the mean; population standard deviation is its nonnegative square root: σ=√[Σ(xᵢ−μ)²/n]. Squaring prevents positive and negative deviations from cancelling. For sample standard deviation, a common convention divides by n−1 instead; use the definition supplied by a calculation question.

The GRE emphasizes interpreting standard deviation and comparing spread rather than elaborate computation. Standard deviation is zero exactly when all observations equal the mean. Values farther from the mean generally contribute more through squared deviations, so an outlier can have a large effect.

07 / UNDERSTAND & APPLY

Shift and scale rules follow from deviations

Adding c to every observation adds c to mean and median, but leaves range and standard deviation unchanged. Multiplying every observation by k multiplies mean by k and standard deviation and range by |k|. A negative multiplier reverses order, but distances remain nonnegative.

Changing a single observation is different from transforming every observation. Adding the same value to the list as a new observation also differs from adding it to each old value. Read the operation precisely.

08 / UNDERSTAND & APPLY

A summary can conceal the change you care about

Moving the largest observation still farther right usually raises the mean and range while leaving the median fixed if its ordered position does not change. Median is more resistant to extreme values, but is not immune to every change in data.

A claim that one set is “more variable” needs a named measure and sufficient information. The same mean, median or range does not fix standard deviation. In QC, construct two data sets satisfying the summaries before deciding whether the comparison is forced.

09 / UNDERSTAND & APPLY

Choose a method and answer the requested question

All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.

A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.

A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.

10 / UNDERSTAND & APPLY

Original practice: totals, positions and spread

When standard deviation must be calculated, the question explicitly supplies the population convention.