LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A critical value is the point on a distribution where the probability to the left equals a level you chose. On the standard normal curve, the bell with mean 0 and variance 1, the lecture asks for the point that leaves 95 percent of the probability to its left. The percent-point function returns about 1.65. Feed 1.65 back into the cumulative function and you get 95 percent again. The density draws the curve. The cumulative function measures area under it. The critical value is the x that makes the area equal your target.
Why it matters
Tests and intervals need a cutoff, not a picture. The lecture’s route into that cutoff is the pair of limit theorems. The law of large numbers pulls a sample average toward the population mean as the sample grows. The central limit theorem says the distribution of that average is approximately normal even when the raw measurements are not. Standardize the average and the reference curve is the standard normal density, the lowercase phi of the lecture. A sample mean of heights, or of any other measurement they mention in the same breath, can be placed on that curve after standardization. The critical value is how you turn “95 percent” into a number on the axis.
Without the inverse step, you can integrate forever and never answer “how far from the mean is far.” The inverse is the whole content of the phrase.
The idea in plain words
The density is the height of the bell. The lecture does not re-derive the formula. It only needs you to know that area under the density is probability. The capital phi, the cumulative distribution function, is that area from negative infinity up to a point x. If a standard normal variable is at most x, the probability of that event is capital phi at x.
Set the area equal to 0.95 and solve for x. That solution is the critical value for a 95 percent left tail. SciPy’s stats.norm.ppf, the percent-point function, is the inverse cumulative function. The lecture calls it on 0.95 and reads about 1.65. The check is the forward direction. The cumulative function at about 1.65 is 0.95. Inverse and forward undo each other. If they do not, you called a different level or a different distribution.
The picture is a vertical line on the bell. Everything to the left of the line is the 95 percent. Everything to the right is the remaining 5 percent. Move the line left and the left area falls. Move it right and the left area rises. The critical value is not a probability. It is the location. 1.65 is not 95 percent. It is the place where the area reaches 95 percent for this particular bell.
Because standardization has already subtracted the mean and divided by the standard deviation, this 1.65 lives in standard units. On the original scale of a height study, you would multiply by the standard error and add the center back. The lecture does not do that arithmetic. It stops at the standard-unit cutoff and at the relationship among density, area, and inverse.
How to use it
Pick the probability first. If you want 95 percent to the left of the cutoff, ask the inverse cumulative function for 0.95 on the standard normal and expect a number near 1.65. Confirm by calling the cumulative function on what you got. The pair of calls is a one-line unit test.
What usually goes wrong
Calling 1.65 a probability confuses the axis with the area. The probability was the input, 0.95. The critical value was the output.
Using the lecture’s 1.65 for a 97.5 percent left tail, or for a two-sided test, imports the right function and the wrong level. The inverse will happily return another point. The error is the level you type, and the cumulative check will catch it if you compare with 0.95 and notice you asked for something else.
Questions people ask
What number did the lecture get for 95 percent?
About 1.65, from the inverse of the standard normal cumulative function at 0.95. Putting that number back into the cumulative function returns 95 percent.
How is that different from the density?
The density is the height of the curve. The cumulative function is the area from negative infinity up to a point. The critical value is the point whose area equals the probability you specified.
Why is the normal curve the reference at all?
Because, in the lecture’s setup, a sample average is approximately normal once samples grow and averages form a bell, and standardization puts that average onto the standard curve where 1.65 is defined.