Difficulty: Intermediate | Prerequisites: basic probability, the normal distribution PDF formula, z-scores, intro to confidence intervals.
The normal distribution is the workhorse of applied statistics, and two of its properties come up constantly: the shape of its probability density function (symmetric, single peak at the mean) and how sample means drawn from it behave. This set of material sits right at the transition from "I know what a normal distribution is" to "I can use it to make inferences about populations." You should already be comfortable with z-scores, the standard normal table, and the formula for the normal PDF. If confidence intervals are new to you, read those notes first.
The normal PDF is symmetric about its mean, peaks there, and drops off as you move away. When you average a sample of n observations, the resulting sample mean is itself normally distributed with a smaller spread (standard error = sigma / sqrt(n)), which is the foundation of confidence intervals and hypothesis testing.
Probability density function (PDF), p_Y(y)
The function that gives the relative likelihood of a continuous random variable taking on a given value. For the normal distribution: p_Y(y) = (1 / (sigma * sqrt(2 * pi))) * exp(-(y - mu)^2 / (2 * sigma^2)).
In simple terms, this is the formula for the bell curve's height at any point y.
Symmetry of the normal distribution
The normal PDF satisfies p_Y(mu - c) = p_Y(mu + c) for any constant c. The curve is a perfect mirror image about its mean.
Think of it as: if you fold the bell curve in half at the mean, the two sides match exactly.
Sample mean (Y-bar)
The arithmetic average of n observations: Y-bar = (1/n) * sum of Y_i. Used as an estimator of the population mean mu.
In simple terms, this is the number you get when you add up all your data and divide by how many you have.
Standard error of the mean
The standard deviation of the sampling distribution of Y-bar: SE = sigma / sqrt(n). It measures how much the sample mean varies from sample to sample.
Think of it as: the bigger your sample, the less your average bounces around.
Sampling distribution
The probability distribution of a statistic (such as Y-bar) computed from all possible samples of size n drawn from a population.
In simple terms, if you could repeat your study thousands of times, the sampling distribution is the pattern of results you would get.
Confidence interval
An interval estimate of a population parameter, constructed so that in repeated sampling, a stated percentage of such intervals would contain the true parameter value. For a normal population with known sigma: Y-bar +/- z* * (sigma / sqrt(n)).
Think of it as: a range of plausible values for the true mean, with a stated level of confidence attached.
Z-score (standard normal value)
The number of standard deviations a value lies from the mean: Z = (Y-bar - mu) / (sigma / sqrt(n)). Used to convert any normal variable to the standard normal (mean 0, variance 1) for probability lookups.
In simple terms, this tells you how "unusual" your result is, measured in units of spread.
Symmetry: p_Y(mu - c) = p_Y(mu + c)
Plug (mu - c) and (mu + c) into the PDF formula. Both give the exponent -(c^2) / (2 * sigma^2), so the outputs are identical.
This holds for any c > 0. The bell curve is symmetric about mu.
The PDF at the mean can exceed 0.5
The PDF value p_Y(mu) is NOT a probability. It is a density, and densities can be greater than 1.
At the mean: p_Y(mu) = 1 / (sigma * sqrt(2 * pi)). If sigma is small enough (e.g. sigma < 1 / sqrt(2 * pi) ≈ 0.399), this value exceeds 1, let alone 0.5.
The statement "p_Y(mu) < 0.5" is therefore false in general. It depends entirely on sigma.
The mean is the global maximum: p_Y(mu) > p_Y(mu + c)
The exponent -(y - mu)^2 / (2 * sigma^2) is maximised when y = mu (the squared term is zero).
For any c > 0, the exponent at (mu + c) is strictly more negative, so the PDF value is strictly smaller.
This confirms the bell curve peaks at the mean and drops off in both directions.
Does p_Y(mu + c) > 2 * p_Y(mu + 2c)?
This asks whether moving one more step of c away from the mean more than halves the density.
Compute the ratio: p_Y(mu + c) / p_Y(mu + 2c) = exp(3c^2 / (2 * sigma^2)).
Since c > 0, this ratio is strictly greater than 1, but it is not necessarily greater than 2 for all c and sigma. For very small c relative to sigma, the ratio approaches 1.
The statement is false in general. It holds for large enough c / sigma, but not universally.
Setting up the sampling distribution
If Y_i are i.i.d. with mean mu and variance sigma^2, then Y-bar has mean mu and variance sigma^2 / n.
When Y_i are normal, Y-bar is exactly normal: Y-bar ~ N(mu, sigma^2 / n).
Standard error: SE(Y-bar) = sigma / sqrt(n). With sigma^2 = 400 and n = 400, SE = 20 / 20 = 1.
Worked example: probability of observing a sample mean (PS4 Q2a)
Suppose mu = 0 (no effect of the tax). We want P(Y-bar <= -1.5).
Standardise: Z = (-1.5 - 0) / 1 = -1.5.
From the standard normal table: P(Z <= -1.5) = 0.0668, so about a 6.68% chance.
Even if the tax has no real effect, you would see a decrease of 1.5 oz or more roughly 1 in 15 times.
Constructing confidence intervals (known sigma)
General formula: Y-bar +/- z* * (sigma / sqrt(n)).
The critical value z* depends on the confidence level. Common values:
70% CI: z* = 1.04
95% CI: z* = 1.96
99% CI: z* = 2.576
Worked example: 70% confidence interval (PS4 Q2b)
Y-bar = -1.5, sigma = 20, n = 400, so SE = 1.
70% CI: -1.5 +/- 1.04 * 1 = (-2.54, -0.46).
Interpretation: we are 70% confident the true mean decrease lies between 0.46 and 2.54 ounces.
Worked example: 99% confidence interval (PS4 Q2c)
99% CI: -1.5 +/- 2.576 * 1 = (-4.076, 1.076).
This interval is much wider, and it includes zero, meaning at 99% confidence we cannot rule out that the tax had no effect.
Choosing sample size for a desired margin of error (PS4 Q2d)
Want the 99% CI to be Y-bar +/- 1 (margin of error = 1 ounce).
Set z* * sigma / sqrt(n) = 1. That gives 2.576 * 20 / sqrt(n) = 1.
Solve: sqrt(n) = 51.52, so n = 2654.3. Round up: n = 2655 people.
Key insight: to cut the margin of error by a factor of k, you need k^2 times as many observations.
p_Y(y) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(y - \mu)^2}{2\sigma^2}\right)\text{SE}(\bar{Y}) = \frac{\sigma}{\sqrt{n}}\text{CI} = \bar{Y} \pm z^* \cdot \frac{\sigma}{\sqrt{n}}n = \left(\frac{z^* \cdot \sigma}{E}\right)^2 \quad \text{(minimum sample size for margin of error } E\text{)}Polling firms use exactly this machinery: they sample n voters, compute a sample proportion, and report a margin of error. The margin of error is the z* * SE piece of the confidence interval formula. When a news broadcast says "the margin of error is plus or minus 3 percentage points," that is a confidence interval at work.
Quality control in manufacturing relies on the same logic. A factory samples n items, measures their dimensions, and builds a confidence interval for the true mean dimension to decide whether the production line is within spec.
Students often think the PDF value at a point is a probability. It is not. A density can exceed 1. Only areas under the curve (integrals) give probabilities.
Students frequently confuse sigma (the population standard deviation) with the standard error sigma / sqrt(n). The standard error is the spread of the sample mean, not of individual observations.
A 95% confidence interval does not mean "there is a 95% probability that mu is in this interval." The interval is fixed once computed. In repeated sampling, 95% of such intervals would contain mu.
Students sometimes think doubling the sample size halves the width of the confidence interval. It does not. Because SE involves sqrt(n), you must quadruple n to halve the width.
⚠️ Proving or disproving statements about the normal PDF using the formula directly is a staple exam question. Know how to substitute into the exponent and simplify.
⚠️ You will almost certainly be asked to construct a confidence interval and interpret it. Have the z* values for 70%, 90%, 95%, and 99% memorised.
⚠️ Sample-size determination questions ("how many people do you need?") appear regularly. Remember to round up, never down.
⚠️ Be precise about the interpretation of a confidence interval. Saying "95% probability mu is in the interval" will lose marks.
True or false: The normal PDF is always less than 1 at the mean. ___
Fill in the blank: The standard error of the sample mean equals sigma divided by ___.
True or false: A 99% confidence interval is narrower than a 95% confidence interval for the same data. ___
Fill in the blank: To cut the margin of error in half, you must multiply the sample size by ___.
True or false: p_Y(mu - 3) = p_Y(mu + 3) for a normal random variable. ___
Answers: 1. False (depends on sigma). 2. sqrt(n). 3. False (wider). 4. Four. 5. True (symmetry).
Q: Let Y ~ N(mu, sigma^2). Prove or disprove: p_Y(mu) > p_Y(mu + c) for c > 0.
A: True. The exponent -(y - mu)^2 / (2 * sigma^2) equals zero at y = mu and is strictly negative at y = mu + c. Since exp(0) > exp(negative number), the PDF at the mean exceeds the PDF at any other point.
Q: If sigma^2 = 400 and n = 400, what is the standard error of Y-bar?
A: SE = sqrt(400) / sqrt(400) = 20 / 20 = 1.
Q: With mu = 0 and SE = 1, find P(Y-bar <= -1.5).
A: Z = (-1.5 - 0) / 1 = -1.5. From the z-table, P(Z <= -1.5) = 0.0668.
Q: Construct a 99% confidence interval for mu if Y-bar = -1.5 and SE = 1.
A: -1.5 +/- 2.576 * 1 = (-4.076, 1.076).
Q: How many people must you sample so that a 99% CI has a margin of error of 1, given sigma = 20?
A: n = (2.576 * 20 / 1)^2 = (51.52)^2 = 2654.3. Round up to n = 2655.
Q: Give a counterexample showing p_Y(mu) < 0.5 is not always true.
A: Let sigma = 0.1. Then p_Y(mu) = 1 / (0.1 * sqrt(2 * pi)) ≈ 3.99, which is far greater than 0.5. The claim fails whenever sigma is small.
This material connects directly to hypothesis testing: the z-score you compute for a confidence interval is the same test statistic you use for a z-test. When a 95% CI excludes zero, you would reject H0: mu = 0 at the 5% level.
The sample-size formula reappears when designing experiments. In clinical trials, power analysis uses the same sigma / sqrt(n) relationship to determine how many subjects are needed to detect a given effect.
The symmetry properties of the normal PDF underpin the entire framework of two-sided tests and two-sided confidence intervals.
Normal distribution, Gaussian distribution, bell curve, PDF, probability density function, sampling distribution, standard error, confidence interval, z-score, z-table, critical value, margin of error, sample size determination, CLT, central limit theorem, sigma, variance, standard deviation, Y-bar, sample mean, population mean, Analytical Statistics, ASTATS, NYU, PS4, Problem Set 4, Timothy Roeper