Shape, center, spread. Three things and you have described any data set.
⭐ The one idea
Raw numbers are noise. A picture shows the shape, one number gives the center, one number gives the spread. Report all three and you have described the data.
💡 Class width = (max − min) ÷ number of classes, rounded up. Classes never overlap and every value lands in exactly one. Relative frequencies add to 1.
A histogram: bars touch, x-axis is numeric classes, height = frequency. Use it for quantitative data.A stem-and-leaf plot keeps every actual value and still shows the shape.
📊 Bar graph
For categorical data. Bars have gaps; order does not matter. Patients per unit, admissions by diagnosis.
🥦 Pie chart
Parts of one whole; slices add to 100%. Fine for 3–6 categories, hopeless for 12.
📈 Histogram
For quantitative data. Bars touch; x-axis is a number line in classes.
📉 Time-series / line graph
One value over time. Vital-sign flowsheet, weekly infection counts.
Pie chart: each slice = category ÷ total.A Pareto chart is just a bar graph sorted tallest to shortest.
🏠 Shape
Symmetric, skewed right (long tail to the right, mean dragged up), skewed left, uniform. Hospital length of stay and costs are almost always skewed right.
🧠 The tail names the skew. Skewed right = tail to the right = a few very large values = mean > median. Income, length of stay, ER wait times. Skewed left: mean < median. Age at death from natural causes.
🎯 Center: mean, median, mode
⚖️ Mean x̄
x̄ = Σx / n = 855 / 8106.875 mg/dL
The balance point. Uses every value, so one outlier drags it.
📌 Median
Sort, take the middle. With 8 values, average the 4th and 5th: (104 + 106) / 2
105 mg/dL
Resistant to outliers. Use it for skewed data (income, length of stay).
🔁 Mode
The most frequent value.
104 mg/dL
The only “center” that works for categorical data (most common blood type: O+).
⚠️ Outlier test with one glance
Change the 126 to 226. Mean jumps to 119.375; median stays 105. When mean and median disagree a lot, the data are skewed or contain an outlier — report the median.
↔️ Spread: range, variance, standard deviation
📏 Range
max − min = 126 − 9234 mg/dL
Quick, but it only looks at two values.
📊 Variance s²
s² = Σ(x − x̄)² / (n − 1)108.4
Average squared distance from the mean. Units are mg/dL² — nobody thinks in those, so take the root.
⭐ Standard deviation s
s = √s²10.4 mg/dL
The typical distance of a value from the mean. Same units as the data. Bigger SD = more spread out.
Divide by n − 1 for a sample (that is what Desmos stdev does). Divide by N only for a whole population (stdevp). Exams almost always want the sample version.
💡 Empirical rule preview: for bell-shaped data about 68% of values sit within 1 SD of the mean. Here that is 106.875 ± 10 → roughly 96 to 117 mg/dL. Full story on the Normal page.
📦 Position: percentiles, quartiles, box plots
Percentile of a value = (number of values below it ÷ n) × 100.
🧪 Second example: length of stay (days) for 10 patients
Outlier fences: Q1 − 1.5·IQR = -1.875 and Q3 + 1.5·IQR = 11.125. The 12-day stay is above the upper fence → outlier.
The box is the middle 50% (Q1 to Q3). The line inside is the median. Whiskers reach the last non-outlier; dots beyond the fences are outliers.
💡 Reading a box plot
Median line off-center → skewed.
Long right whisker → skewed right.
Wider box → bigger IQR → more spread.
Two box plots side by side = the fastest way to compare groups.
🧠 Five-number summary
min, Q1, median, Q3, max. Desmos gives all five with quartile(L,0) … quartile(L,4), or just type boxplot(L) and hover.
⚠️ Different books (and Desmos) split quartiles slightly differently, so your Q1 may differ from a classmate’s by a little. Use the method your instructor showed; on the final, Desmos’s answer is the one that matches the key.
❓ Five patients’ pain scores: 2, 3, 3, 8, 9. Mean? Median? Which describes the typical patient better?
Mean = 25/5 = 5. Median = 3. The two 8-and-9 scores pull the mean up; the median (3) is the better “typical” value here.
❓ Unit A heart rates have SD 6 bpm; Unit B has SD 14 bpm. Which unit’s patients are more alike?
Unit A. Smaller SD = values cluster closer to the mean.
❓ A histogram of ER wait times has a long tail to the right. Which is larger, the mean or the median?
Mean. Right skew drags the mean toward the tail.
MAT 300 · built from your own course files (the statistics study guide, the Desmos guide, the formula sheet and the final-exam study questions) with nursing examples. Not a substitute for the textbook — check any number against your own notes before an exam.