lesson 5: STANDARD DEVIATION AND Z-SCORES

standard deviation (SD)

- tells us how spread out the

scores are from the mean.

- is the square root of the average squared deviation from the mean (variance).

small SD

means most scores are close to the mean

large SD

means scores are widely scattered.

Estimated Standard Deviation

is your best guess as to the standard deviation of a population of scores based on information known about only

a small subset or sample of scores from that population.

Variance

- Average of the squared

deviations

- Expressed in squared units

- Less intuitive

- More sensitive to outliers;

amplifies the effect of outliers

z-score

- tells us how many standard deviations away a score is from the mean.

- Combining information from different measures that are on different scales.

Sign (positive or negative)

tells us which half of the

distribution the z-score falls

Magnitude (actual number)

tells us, in units of SD, how

far away the score is from the center or mean (generally

fall between -3 and 3).

Normal Distributions

- There are a lot of normal

distributions!

- Bell curve; "Gaussian Curve"

- can differ in

their means and their standard

deviations.

- are similar in their shape and the proportion of scores within a given distance along the x-axis.

68-95-99.7 Rule

~68% of data lies within 1 SD

~95% within 2 SDs

~99.7% within 3 SDs

Area under the Curve

The total area under the normal curve = 1.0 (100%)

Between z = -1 and z = +1 → ~68% of scores

Outside z = ±1 → 32% of scores (16% in each tail)

Between z = -2 and z = +2 → ~95% of scores

Beyond ±2 SD → only 5% of scores (rare cases)

Symmetric around the mean.

Mean = Median = Mode.

Total area under the curve = 1.0 (or 100%).

Most scores cluster around the center.

Defined by mean (μ) and SD (σ).

68% of scores are within 1 SD of the mean.

95% within 2 SDs, and 99.7% within 3 SDs.

(This is called the 68-95-99.7 Rule).

7 Features of Normal Distributions: