6 mistakes learners make with the normal distribution, each one named and explained.
Standardise first, then read the tail the question actually asked for.
The standard error is given as sigma over n.
Dividing by the sample size is the shape of the formula and the root is the small detail.
Question. Standard error
A common answer. sigma / n
The answer. sigma / root(n)
Why. The variance divides by n, and the standard deviation is its square root.
Take the root of the variance formula. The n comes out as root n.
Sigma is given as the standard deviation of the sample mean.
Sigma is the standard deviation in the question, so it is the one to hand.
Question. Standard error of the sample mean
A common answer. sigma
The answer. sigma over the root of n
Why. Averaging several observations makes the average less variable than any single one.
A mean of many is always steadier than one value. The standard error has to be smaller.
A probability is read off without converting the value to a z-score first.
The numbers in the question look usable and standardising is an extra step.
Question. Mean 50, standard deviation 8, P(X < 58)
A common answer. 0.500
The answer. 0.841
Why. 58 is one standard deviation above the mean, which is z = 1.
Every normal question starts by turning the value into a z. Nothing works before that.
Sigma squared over n is offered as the standard error.
The variance formula is the one derived first and it is the more memorable expression.
Question. Standard deviation of the sample mean
A common answer. sigma^2 / n
The answer. sigma / root(n)
Why. Sigma squared over n is the variance; the standard deviation is its root.
Check the units. A standard deviation always matches the data it describes.
P(X < 58) is answered with the upper tail probability, or a value is found on the wrong side.
Tables and calculators give one tail by default, and which one is easy to lose track of.
Question. Mean 50, standard deviation 8, find P(X < 58)
A common answer. 0.159
The answer. 0.841
Why. 58 is above the mean, so more than half the distribution lies below it.
Sketch the curve and shade the region. The shaded fraction tells you if the answer is sensible.
k is found as the mean plus the z-value, with the standard deviation never applied.
The z-value looks like a number of units, because in standardised terms it is.
Question. Mean 70, standard deviation 5, z = 1.2816
A common answer. 70 + 1.2816 = 71.3
The answer. 70 + 1.2816 x 5 = 76.4
Why. A z-value counts standard deviations, so it has to be multiplied by one.
Rearranging z = (x - mean) / sd always leaves the standard deviation as a multiplier.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.