Density Curves, Percentiles & Cumulative Plots
- Interpret a density curve as a model in which area represents proportion
- Read percentiles from a cumulative relative frequency plot
- Distinguish a percentile from a z-score and from a raw score
A density curve is a model, and area is proportion
A density curve is an idealized smooth description of a distribution. Its defining properties are that it is never negative and that the total area beneath it is exactly 1. The area under the curve between two values is the proportion of observations falling in that interval. This is what makes the Normal curve usable: once you accept the model, every probability question becomes an area question, and the table or calculator simply reports areas.
The mean and median on a density curve
The median of a density curve is the value that splits the area in half. The mean is the balance point — where the curve would balance if it were a solid shape of uniform thickness. For a symmetric curve the two coincide. For a right-skewed curve the long tail pulls the balance point to the right of the halfway point, so the mean exceeds the median; for left skew the reverse. Knowing which is larger from the shape alone is a standard multiple-choice item.
Percentiles and cumulative relative frequency plots
The p-th percentile is the value below which p percent of observations fall. A cumulative relative frequency plot (an ogive) puts the variable on the horizontal axis and the accumulated proportion on the vertical, so it rises from 0 to 1 and never falls. To read a percentile, enter on the vertical axis at the proportion and read across to the curve and down; to find the percentile of a value, do the reverse. The plot is steepest where the data are densest, which is why an ogive of a symmetric distribution is S-shaped.
Keep three quantities distinct. The raw score is the measurement in its own units. The z-score says how many standard deviations from the mean it lies and is unitless. The percentile says what proportion of the distribution lies below it. They are three different answers to three different questions, and a question asking for one earns nothing for another.
Scores are approximately Normal with mean 500 and standard deviation 100. A student scores 640. Find the z-score and describe the percentile. Separately, find the score at the 90th percentile.
- 1.z = (640 − 500)/100 = 1.40, so the score is 1.4 standard deviations above the mean.
- 2.The area to the left of z = 1.40 under the standard Normal curve is about 0.919, so the score is at roughly the 92nd percentile.
- 3.For the 90th percentile, find the z with 0.90 of the area below it: z ≈ 1.28.
- 4.Convert back: x = μ + zσ = 500 + 1.28(100) = 628.
For a right-skewed density curve, which is true?
On a cumulative relative frequency plot, the steepest section of the curve corresponds to —
Answer the 2 checkpoints as you read.
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