A webR tutorial

Explore your own normal distribution

A decorative image of a normal distribution.

Explore Your Own Normal Distribution

Think of a continuous variable you’re interested in that might be normally distributed. Some examples:

  • SAT scores (mean ≈ 1050, sd ≈ 200)

  • Daily coffee consumption in ounces (mean ≈ 12, sd ≈ 4)

  • Reaction time in milliseconds (mean ≈ 250, sd ≈ 50)

  • Daily steps for college students (mean ≈ 8000, sd ≈ 2000)

  • Hours of sleep per night (mean ≈ 7.5, sd ≈ 1.2)

Modify the code below to explore your variable of interest. Click Run Code to see the result. Then use the prompts below the graph for further exploration/consideration.

Explore and consider

1) Translate vs. spread

  • Hold my_sd fixed; change my_mean.
    → Notice the curve shifts left/right but doesn’t change shape.

  • Hold my_mean fixed; change my_sd.
    → Bigger sd = wider & lower peak (area stays 1). Smaller sd = narrow & tall.

2) Middle mass (area = probability)

  • Compare the printed “middle 95%” interval to the plot.
    → Is it symmetric around the mean? (It should be ~mean ± 1.96*sd.)

  • Try sd values that make the 95% range unrealistic for your variable — what does that tell you about a reasonable sd?

3) The 68–95–99.7 rule

  • Estimate by eye where mean ± sd, mean ± 2*sd, mean ± 3*sd fall.

  • How much of the curve seems to lie within each band? (≈ 68%, 95%, 99.7%.)

4) Percentiles & cutoffs (quantiles)

  • Find the value for the top 10%: qnorm(p = 0.90, mean = my_mean, sd = my_sd).

  • Find the median and quartiles: qnorm(p = c(0.25, 0.5, 0.75), mean = my_mean, sd = my_sd).

5) Tail probabilities (single-sided & two-sided)

  • “What’s the chance a draw is above X?” → 1 - pnorm(q = X, mean = my_mean, sd = my_sd).

  • “Between A and B?” → pnorm(q = B, mean = my_mean, sd = my_sd) - pnorm(q = A, mean = my_mean, sd = my_sd).

6) Plausible domain

  • If your variable can’t be negative (e.g., hours, steps), does your chosen sd put non-trivial mass below 0?
    → If yes, consider a smaller sd or note that a normal model may be imperfect at the lower tail.