Descriptive Statistics

01 / UNDERSTAND & APPLY

1. A summary describes a specified collection

Descriptive statistics compress a data collection into quantities such as its center and spread. Before calculating, identify the observational unit, the variable, its units and the number of observations. A list of measurements and a frequency table may represent the same data, but a table of class intervals usually contains less information than the original list.

A numerical summary does not replace the distribution. Two data sets can have the same mean and different spreads, or the same mean and variance with different shapes. A sensible interpretation states what the summary captures and what it leaves unresolved, especially when outliers or multiple clusters are present.

For observations x₁,…,xₙ with n≥1, the arithmetic mean is x̄=(Σxᵢ)/n. With distinct values xⱼ occurring at frequencies fⱼ, use n=Σfⱼ and x̄=(Σfⱼxⱼ)/n. Frequencies count repeated observations; treating each distinct value as equally frequent changes the data.

A weighted mean Σwᵢxᵢ/Σwᵢ uses nonnegative weights with positive total weight. It lies between the smallest and largest values receiving positive weight. If the weights represent group sizes, it is the combined mean. If they represent something else, explain that meaning rather than automatically calling the result an ordinary unweighted average.

02 / UNDERSTAND & APPLY

2. Different centers answer different questions

The median is determined after sorting the observations. For odd n it is the middle observation, at position (n+1)/2. For even n the usual convention is the average of the two central observations, at positions n/2 and n/2+1. The median need not itself occur in the data when n is even.

The mode is a most frequent value. There may be more than one mode. When every value occurs equally often, conventions vary between calling every value modal and reporting no unique mode; state the convention rather than implying that every data set has exactly one. A modal category can be meaningful even when numerical means are not.

The mean responds to the magnitude of every observation. Changing one value by d changes the mean by d/n. The median depends on ordering and middle positions, so an already extreme value can move further outward without changing it. This resistance does not mean the median is unaffected by every possible data change.

For a strongly skewed distribution, mean and median can differ substantially. There is no universal exact equation tying mean, median and mode together. Any empirical approximation between them is a contextual rule of thumb, not an algebraic identity that may be imposed on arbitrary data.

03 / UNDERSTAND & APPLY

3. Spread needs both a center and a convention

The range is maximum minus minimum. It is simple but depends only on two observations. Mean absolute deviation about a chosen center c is (1/n)Σ|xᵢ−c|; the chosen center must be stated. Mean deviation about the mean and mean deviation about the median need not have the same value.

The population-style variance of a finite data list is σ²=(1/n)Σ(xᵢ−x̄)². It is nonnegative and equals zero exactly when all observations are identical. Standard deviation σ is its nonnegative square root. If the observations are measured in centimeters, variance is in square centimeters and standard deviation is in centimeters.

Expanding the squares gives Σ(xᵢ−x̄)²=Σxᵢ²−n x̄², since Σxᵢ=n x̄. Therefore σ²=(Σxᵢ²)/n−x̄². This raw-moment formula is algebraically convenient, though direct subtraction of nearly equal large rounded numbers can be numerically unstable.

When estimating a population variance from a random sample, the usual sample variance is s²=Σ(xᵢ−x̄)²/(n−1), for n≥2. It differs from the descriptive divisor-n variance by n/(n−1). The unbiasedness motivation uses a sampling model; it is not a reason to change the denominator silently in a deterministic data-summary question. This lesson states which convention each problem uses.

04 / UNDERSTAND & APPLY

4. Centering removes the cross term

For any real center c, write xᵢ−c=(xᵢ−x̄)+(x̄−c). Squaring and summing produces a cross term containing Σ(xᵢ−x̄), which is zero. The result is Σ(xᵢ−c)²=Σ(xᵢ−x̄)²+n(x̄−c)².

The final term is nonnegative and vanishes only when c=x̄. Thus the arithmetic mean uniquely minimizes the sum of squared deviations for a nonempty finite data set. This identity explains why variance is centered at the mean and why using an assumed mean requires a correction term.

For an assumed center a, let dᵢ=xᵢ−a. Then x̄=a+(Σdᵢ)/n and σ²=(Σdᵢ²)/n−((Σdᵢ)/n)². Merely averaging dᵢ² gives deviations about a, not the variance unless a equals the actual mean.

Absolute deviations behave differently: a median minimizes Σ|xᵢ−c|. Pairing the smallest and largest observations shows why: their total distance to c is minimal whenever c lies between them. Intersecting the central intervals gives the middle observation for odd n or the interval between the central two for even n. The conventional even-sample median is one minimizer among that whole interval.

05 / UNDERSTAND & APPLY

5. A shift changes location; a scale changes spread

If each observation is transformed by yᵢ=axᵢ+b, then ȳ=a x̄+b by linearity of the sum. Also yᵢ−ȳ=a(xᵢ−x̄), so Var(y)=a²Var(x) and SD(y)=|a|SD(x). The absolute value is essential because standard deviation cannot be negative.

Adding a constant changes the mean and median by that constant but leaves variance, standard deviation and range unchanged. Multiplying by a negative constant reverses the order of observations while multiplying the range and standard deviation by its absolute value. The conventional median still transforms as a·median+b.

The same variance scaling holds for sample variance because the common divisor n−1 is unchanged. Mean absolute deviation about the correspondingly transformed center scales by |a|. These rules can replace long recalculations, but only when the same affine transformation is applied to every observation.

The coefficient of variation is commonly CV=100·σ/x̄ percent when x̄>0 on a meaningful ratio scale. It compares relative spread. It is unchanged by multiplication by a positive constant, but not by translation. Near-zero means or arbitrary-origin scales make this ratio unstable or inappropriate; do not use it as a universal ranking device.

06 / UNDERSTAND & APPLY

6. Overall spread includes differences between group means

For groups of sizes n₁,n₂ with means μ₁,μ₂, the combined mean is μ=(n₁μ₁+n₂μ₂)/(n₁+n₂). Averaging the two means without weights is valid only when the group sizes are equal or when the values happen to make the weighting irrelevant.

If σ₁²,σ₂² are divisor-n variances within the groups, the combined variance is [n₁(σ₁²+(μ₁−μ)²)+n₂(σ₂²+(μ₂−μ)²)]/(n₁+n₂). Apply the centering identity within each group to derive it. The first part measures within-group scatter; the second measures separation of the group centers.

A weighted average of within-group variances alone omits the between-group contribution. Even if every observation within each group is identical, the combined variance can be positive when the group means differ. This is why pooling spread requires more than pooling the variance numbers.

If sample variances are supplied instead, first recover each within-group squared-deviation total as (nⱼ−1)sⱼ². Add the between-group terms and then divide by N or N−1 according to the requested combined convention. Mixing n and n−1 formulas without conversion is a common source of error.

07 / UNDERSTAND & APPLY

7. Correct totals before recomputing statistics

If one recorded value w should have been c, the corrected sum is S−w+c and the corrected squared sum is Q−w²+c². Both totals change. Correcting only the mean and leaving the old squared total produces an inconsistent variance.

For a data set of known size n, mean μ and divisor-n variance σ², recover S=nμ and Q=n(σ²+μ²). These totals are sufficient to recompute mean and variance after replacing, adding or removing known observations. The new denominator changes when an observation is added or removed.

Missing observations can often be found from a sum and a squared sum. If two unknown values have sum s and squared sum q, their product is (s²−q)/2. They are roots of t²−st+(s²−q)/2=0. A negative discriminant means no real values satisfy the reported summaries.

The final result should be checked for nonnegative variance and any constraints on the original data, such as nonnegativity or integrality. Rounded reported means and variances may not determine exact missing observations; distinguish exact summaries from rounded measurements.

08 / UNDERSTAND & APPLY

8. Interval summaries require explicit approximations

When only class intervals and frequencies are known, the individual observations are not. Replacing each observation by its class midpoint gives an estimated mean and variance. These are exact for the midpoint-representative data set, but generally approximate for the unknown original data.

For a continuous grouped distribution, a common median estimate is L+h(N/2−C)/f, where L is the lower boundary of the median class, h its width, C the cumulative frequency before it and f its frequency. This linearly interpolates the cumulative count within that class, effectively assuming a uniform distribution there. The class boundaries must be consistent and non-overlapping.

For equal-width classes with a unique modal class and neighboring frequencies f₀,f₁,f₂, a conventional grouped mode estimate is L+h(f₁−f₀)/(2f₁−f₀−f₂). It is a histogram-based interpolation, not an exact recovery of the original most frequent value. Unequal widths require density-aware treatment; simply choosing the largest raw class frequency can be misleading.

Quartiles and percentiles split ordered data by position, but several interpolation conventions exist for finite samples. State the convention before computing. The interquartile range Q₃−Q₁ measures the middle spread. A 1.5-IQR fence is a screening rule for unusual observations, not proof that a point is erroneous or must be deleted.

09 / UNDERSTAND & APPLY

9. Original practice with full solutions

Unless a question explicitly requests sample variance, use the divisor-n descriptive convention. Retain exact fractions until the final step.

10 / UNDERSTAND & APPLY

10. Audit the denominator, units and information loss

A complete statistical calculation states the data count, the weighting and the variance convention. Sorting is necessary for positional summaries, while frequencies must be respected for weighted sums. A variance cannot be negative, and a mean with nonnegative weights cannot lie outside the observed range.

Use transformations and raw totals to avoid unnecessary repetition. A common shift leaves spread unchanged; a common scale changes variance by its square and standard deviation by its absolute value. Combined-group variance includes both internal variation and separation of the means.

For grouped data, call midpoint calculations estimates and explain interpolation assumptions. For quartiles, identify the chosen convention. Do not turn an outlier flag into a claim of error, or an empirical mean-median-mode approximation into an exact theorem.

Finally, interpret the number in the original units and context. A center without spread may conceal a widely scattered distribution, and spread without the center may conceal a scale difference. When two summaries seem contradictory, return to the observations, the denominator and the definition before changing arithmetic.