Descriptive Statistics & Sampling

The mean sums every value, so a long right tail of very high incomes pulls it upward. The median depends only on rank, so extreme values barely move it. Mean above median is the usual signature of right skew.

Nominal values are labels with no order. Ordinal values are ranked but without known equal gaps. Interval values have equal gaps and an arbitrary zero. Ratio values have equal gaps and a true zero, so ratios are meaningful.

Variance is the average squared deviation from the mean, so it is measured in squared units. Standard deviation is the square root of the variance, which puts the number back into the data's original units and makes it readable.

Divide by N when your numbers are the whole population you describe. Divide by n-1 when they are a sample standing in for a larger population's spread. The n-1 version is always the larger of the two.

Averaging rates weights every team equally regardless of size, so a tiny team counts as much as a huge one. The correct overall rate is total conversions over total users — a size-weighted mean.

The mean adds every value in full, so one enormous salary drags the average up. The median depends only on which value sits in the middle position, so an extreme point changes the ordering but barely moves the centre.

The five-number summary is the minimum, first quartile, median, third quartile and maximum. Together they show where the bulk of the data sits and how far each tail reaches, which a single average cannot express.

Positive sample skewness means the long tail stretches to the right, toward large values. Negative means the long tail stretches to the left. A value near zero indicates a roughly symmetric shape, like a normal bell.

It takes the first and third quartiles Q1 and Q3, sets IQR = Q3 - Q1, and marks any point below Q1 - 1.5 x IQR or above Q3 + 1.5 x IQR. Those cutoffs are called the fences.

Mean latency averages away the slow requests that users actually notice. The 95th and 99th percentiles report the experience of the worst 5 percent and 1 percent of requests, which is where dissatisfaction, timeouts and retries live.

A joint proportion divides a cell count by the grand total. A marginal proportion divides a row or column total by the grand total. A conditional proportion divides a cell by its own row or column total.

A correlation only says the two series move up and down together in the observed data. It cannot say ice cream causes drownings: something else, such as hot weather, can drive both, and correlation carries no direction while causation does.

Sample covariance measures whether two variables move together, but it carries the product of their units, so its size means little alone. Pearson's r divides covariance by both standard deviations, giving a unitless number between -1 and 1.

Regression to the mean is the tendency for an extreme measurement to be followed by a less extreme one on remeasurement. Expect it whenever two measurements are correlated but not perfectly, because chance helped produce the extreme.

Spearman's rho is a correlation computed on the ranks of the data instead of the raw values. It measures how well two variables move together in a consistently increasing or decreasing pattern, not just along a straight line.

Simple random sampling gives every unit in the frame an equal chance of selection. Stratified sampling first splits the population into non-overlapping groups, then samples inside each one, guaranteeing every group appears and usually giving a more precise estimate.

MCAR means missingness is unrelated to any variable. MAR means it depends only on variables you observed. MNAR means it depends on the missing value itself, as when the highest earners are the ones who leave salary blank.

Badly performing funds get closed or merged away, so a table of funds still open today lists mostly winners. Averaging it measures the survivors, not the return an investor could have expected when picking a fund years ago.

Every filled value sits exactly at the mean, so it adds nothing to the sum of squared deviations while still adding to the row count. The numerator is unchanged and the denominator grows, so variance and standard deviation shrink.

Two selection failures compounded. The mailing list was built from telephone directories, car registrations and subscriber rolls, which excluded poorer households; and only about a quarter of recipients mailed a ballot back, and returners differed from non-returners.

Check whether 8% is inside the normal spread of Tuesday-over-Tuesday changes for this metric, then rule out calendar effects and a tracking break. Only a delta outside normal variation, measured on clean data, is worth investigating as a product problem.

Reconcile the metric against an independent source of truth, such as the records the product itself writes. A tracking break usually starts abruptly at a release, hits only one platform or app version, and leaves the downstream business records unchanged.

Pick a baseline of stable weeks, take a centre line from them, and estimate the spread from consecutive-week differences. Set limits at the centre plus and minus about three spreads; weeks outside get investigated. Exclude known-abnormal weeks from the baseline.

Ask how much of the aggregate drop that cell can explain, and whether an extreme cell was expected anyway. Scanning 40 slices guarantees a few look alarming, and small cells swing hardest, so size the contribution first.

Revenue is usually concentrated in a few large accounts, so check whether one or two explain the whole 6%. Recompute the delta excluding the top accounts: if it vanishes, this is an account event, not a broad product change.

Break the target number into a chain of estimable factors: city population, people per household, share of households owning a piano, tunings per piano per year, and tunings one tuner performs per year. Multiply through, divide, and land near 50.

Write the unit beside every factor and check that they cancel to leave dollars per year: students times participation gives meals per day, times school days per year gives meals per year, times dollars per meal gives the answer.

Bottom-up builds from unit economics: stores times cups per store per day times price per cup times days per year. Top-down starts from the whole out-of-home coffee market and takes the chain's share. Run both and reconcile the gap.

Rank factors by the width of their plausible range as a ratio, not by their size. In a product relative errors add, so a factor uncertain by 3x dominates one known to 10 percent. Halve and double it.

An estimate is good enough when the decision does not flip anywhere across its plausible range. Write down the threshold that would change the call first, then check whether the low-to-high band straddles it.

ARPU is revenue over active users, so pull each part's percent change separately: the ARPU growth factor equals the revenue growth factor divided by the user growth factor. A drop while revenue still rises means the user base grew faster than revenue.

The overall rate is a traffic-weighted average of the channel rates. If traffic shifts toward a low-converting channel, that weighted average can fall even when every channel improves. This is a mix shift, not a conversion regression.

It applies both populations' age-specific rates to one shared reference age distribution, so the comparison no longer reflects the fact that one population is older. The output is a hypothetical rate built for comparison, never either population's real death count.

Partition the total into mutually exclusive, exhaustive segments, compute each segment's absolute delta, and reconcile the bars to the headline. Report gross gains and gross losses separately, not just the net, and give entrants and exits their own bars.

Report the raw number as the headline — it is what the business actually experienced — with the mix-adjusted figure beside it as the explanation. Never substitute one for the other, and never choose the standard mix after seeing the results.