23.1 Exploring & Describing Data

Why Exploring Data Comes First

The PA-CAT (Physician Assistant College Admission Test), developed by Exam Master Corp., dedicates roughly 5% of its 240 items to Statistics — about 12 questions across the four blueprint groups listed in Bulletin Table 11 (PA-CAT Bulletin of Information, rev. 20240815): Exploring Data, Anticipating Patterns, Sampling & Experimentation, and Statistical Inference. Exploring Data is the entry point because every later inference depends on correctly classifying variables, choosing the right display, and summarizing the distribution honestly. On the PA-CAT you will rarely be asked to compute a regression line by hand; instead you must interpret a graph, choose the resistant statistic, or identify what a skew implies for the mean versus the median.

Data Types: Classify Before You Calculate

The first decision in any statistics problem is what kind of variable am I dealing with? The answer dictates every downstream choice.

A common PA-CAT trap is treating an ordinal variable (pain score, NYHA class) as if it were interval-ratio. A change from pain 2 to 3 is not guaranteed to equal a change from 8 to 9, so reporting a mean pain score without acknowledging the ordinal nature is questionable. Conversely, discrete counts such as "number of ED visits per month" can still have a roughly normal distribution when the mean is large, which matters for later inference.

Displays That Match the Data

The PA-CAT expects you to pair each variable type with the correct graph.

• Bar chart — categorical counts; bars have gaps; order is arbitrary for nominal.
• Pie chart — categorical proportions when the whole adds to 100%; avoid if >5 slices.
• Histogram — quantitative distribution; bars touch; bin width changes the story.
• Stem-and-leaf plot — small quantitative data sets; preserves raw values and shape.
• Boxplot (box-and-whisker) — five-number summary; ideal for side-by-side group comparisons.
• Scatterplot — two quantitative variables; the only honest first look at bivariate data.

A histogram shows shape, center, and spread of one quantitative variable, while a boxplot summarizes the same distribution with the five-number summary (min, Q1, median, Q3, max) and flags outliers explicitly. Stem-and-leaf plots are rare on screen but can appear on the PA-CAT as a quick way to display, say, 20 resting heart rates; the trick is that the "stem" is the leading digit and the "leaf" is the trailing digit, so you can reconstruct every value.

Measures of Center

The three measures of center answer different questions.

For a symmetric distribution the mean and median coincide. For a right-skewed distribution (typical of hospital charges, lengths of stay, income) the mean is pulled in the direction of the tail and exceeds the median. For a left-skewed distribution (e.g., age at a retirement clinic) the mean is less than the median. On the PA-CAT, if you read that a distribution of charges is right-skewed and the question asks which is larger, answer mean > median, and choose the median if asked for the "typical" value.

Measures of Spread

• Range = max − min; simple but inflated by outliers.
• Interquartile range (IQR) = Q3 − Q1; resistant, used in boxplots.
• Variance = average squared deviation from the mean; units are squared.
• Standard deviation (SD) = √variance; same units as the data.

A vital identity: SD ≈ range/4 as a rough check for roughly bell-shaped data (the range rule of thumb). If a data set of systolic blood pressures ranges 90–170, a reasonable SD estimate is 20 mmHg. The PA-CAT may give you a tiny data set and ask you to compute the SD; remember to divide by n−1 for a sample, not n.

Position: Percentiles and z-Scores

A percentile tells you what percentage of observations fall at or below a value. A z-score standardizes position: z = (x − mean) / SD. A z of +1.5 means the value is 1.5 standard deviations above the mean. z-scores let you compare a patient on two different scales (e.g., a creatinine of 1.4 versus a BUN of 30) by expressing both in standard-deviation units.

Outliers and the 1.5×IQR Rule

A value is flagged as an outlier when it falls below Q1 − 1.5×IQR or above Q3 + 1.5×IQR. On a boxplot these appear as individual dots beyond the whiskers. In clinical data an outlier may be a data error (a height recorded as 720 cm) or a genuine extreme (a 12-hour ED wait). The PA-CAT tests whether you know not to discard an outlier silently — investigate first, and use resistant statistics if the outlier is real.

Bivariate Data, Scatterplots, and Correlation

A scatterplot plots each subject on two quantitative axes. The correlation coefficient r ranges from −1 (perfect negative linear) to +1 (perfect positive linear); r = 0 means no linear relationship, not no relationship at all. A classic exam trap: a curved (quadratic) scatterplot can have r ≈ 0 even though the variables are strongly related. Three rules to keep straight:

1. Correlation is not causation. Ice cream sales and drowning incidents correlate because of a confounder (summer heat), not because one causes the other.
2. Correlation is sensitive to outliers. A single influential point can manufacture or erase r.
3. Correlation has no units. Switching pounds to kilograms does not change r.

When you describe a scatterplot on the PA-CAT, report direction (positive/negative), form (linear/curved/clusters), strength (how tightly points follow the form), and unusual features (outliers, influential points). The least-squares regression line y-hat = b0 + b1·x minimizes the sum of squared residuals, where a residual = observed y − predicted y. The slope b1 = r·(Sy/Sx). A residual plot with no pattern confirms linearity; a funnel shape signals non-constant variance. These bivariate tools link directly to inference later in the chapter.