● The standard deviation will be in the same units

as the dependent variable.

● The mean and the standard deviation will be

independent.

● Everything you need to know about the

distribution (as a whole) can be summed up with

these two descriptive statistics.

● The standard deviation gives the average amount

that a single score in the distribution deviates

from the mean.

● The normal distribution is unimodal, symmetric and has two inflection points.

● Normally distributed scores are systematically

related to the standard deviation.

Why is the normal distribution important?

● The normal is common in nature; many natural

phenomena are distributed normally.

● The normal is well behaved; its shape and

probability density are predictable.

● The normal is algebraically convenient; sums and

differences of normals are meaningful.

● The normal has a unique property: the mean and

standard deviation are independent.

● If a variable is distributed normally, then there will

be a clear and systematic relationship between

samples and the underlying population.

The Central Limit Theorem

● The central limit theorem explains why many of thedistributions that we work with in Psychology are distributed normally.

● The Central Limit Theorem states:

● The sum of many random effects is distributed

normally, even if the distributions of the

individual random effects are not normally

distributed.

When are distributions in nature not normal?

● When there are ceiling or floor effects on the

measurement.

● When the variable is measured using a nominal or

ordinal scale.

● When the data are sampled from a non-stationary

process (e.g., if the samples are growing).

● When the data are governed by a single, direct

random effect that is not normal (e.g., the roll of a

single die).

z scores

● It is impossible to interpret a single score drawn

from a normal distribution unless you know the

mean and standard deviation of the distribution.

● By converting the raw score to a z score we can

encapsulate all necessary information about the

relationship of the specific score to the mean and

SD of the entire sample.

● If the data are distributed normally then the

distribution of z scores will have a mean of 0 and a

standard deviation of 1.

● A z score is only meaningful if the data are

distributed normally!

● If this is not true then the standard deviation

won't be related to the percentage of scores

under the distribution and the z scores will be

meaningless.

Coefficient of variation

● The coefficient of variation is a “unitless” statistic

that describes the spread of the data relative to

the mean.

● It is only appropriate if the data were measured

on a ratio scale.

Skew

Skew refers to the (lack of) symmetry of a

distribution.

Kurtosis

● Kurtosis refers to the concentration of the data in

a distribution around the mean.

● Leptokurtotic (High)

● Platykurtotic (Low)

Two ways to describe a distribution

● Specify the probability of occurrence of each bin.

The accuracy of this description will scale as the

square root of n.

● Specify all the moments. The first moment will be

most accurate, the higher moments less so.

Why should we bother with moments?

● The mean and the variance (and the higher

moments, for many non-normal distributions)

provide a much more efficient description of the

data than a list of all of the individual observations.

● The mean and the variance are the descriptive

statistics that are used in many inferential tests

that compare distributions across experimental

conditions (or different groups).