a.k.a. “bell curve”

If a characteristic is normally distributed in a population, the distribution of scores measuring that characteristic will form a bell-shaped curve.

This assumes every member of the population possesses some of the characteristic, though in differing degrees. examples: height, intelligence, self

esteem, blood pressure, marital satisfaction, etc.

Researchers presume that scores on most variables are distributed in a “normal” fashion, unless shown to be otherwise Including communication variables

Only interval or ratio level data can be graphed as a distribution of scores: Examples: physiological

measures, ratings on a scale, height, weight, age, etc.

Any data that can be plotted on a histogram

Nominal and ordinal level data cannot be graphed to show a distribution of scores nominal data is usually shown

on a frequency table, pie chart, or bar chart

Lower scores are found toward the left-hand side of the curve.

Medium scores occupy the middle portion of the curve this is where most scores

congregate, since more people are average or typical than not

Higher scores are found toward the right-hand side of the curve

In theory, the “tails” of the curve extend to infinity (e.g. asymptotic) lower

scoresmediumscores

higherscores

In a normal distribution, the center point is the exact middle of the distribution (the “balance point”)

In a normal, symmetrical distribution, the mean, median, and mode all occupy the same place

meanmedianmode

Note the height of the curve does not reflect the size of the mean, but rather the number of scores congregated about the mean

Kurtosis refers to how “flat” or “peaked” a distribution is.

In a “flat” distribution scores are spread out farther from the mean There is more variability

in scores, and a higher standard deviation

In a “peaked” distribution scores are bunched closer to the mean There is less variability

in scores, and a lower standard deviation

kurtosis

Non-normal distributions may be:

Leptokurtic (or peaked) Scores are clustered

closer to the mean Mesokurtic (normal, bell

shaped) Platykurtic (flat)

Scored are spread out farther from the mean

Skewness refers to how nonsymmetrical or “lop-sided” a distribution is.

If the tail extends toward the right, a distribution is positively skewed

If the tail extends toward the left, a distribution is negatively skewed

skewness

In a positively skewed distribution, the mean is larger than the median

In a negatively skewed distribution, the mean is smaller than the median

Thus, if you know the mean and median of a distribution, you can tell if it is skewed, and “guesstimate” how much.

Only 2% of Americans earned more than $250,000 per year in 2005

Statisticians have calculated the proportion of the scores that fall into any specific region of the curve

For instance, 50% of the scores are at or below the mean, and 50% of the scores are at or above the mean

50% 50%

Statisticians have designated different regions of the curve, based on the number of standard deviations from the mean

Each standard deviation represents a different proportion of the total area under the curve

Most scores or observations (approx. 68%) fall within +/- one standard deviation from the mean

-1 SD-2 SD-3 SD +1 SD +2 SD +3 SD

34.13%34.13%

68.26%

Thus, the odds of a particular score, or set of scores, falling within a particular region are equal to the percentage of the total area occupied by that region -1 SD-2 SD-3 SD +1 SD +2 SD +3 SD

68.26%

95.44%

99.72%%

13.59%

34.13%

2.14%

13.59%

34.13%

2.14%

68.2% of all scores should lie within 1 SD of the mean

95.4% of all scores should fall within 2 SDs of the mean

99.7% of all scores should fall within 3 SDs of the mean

The odds that a score or measurement taken at random will fall in a specific region of the curve are the same as the percentage of the area represented by that region.

Example: The odds that a score taken at random will fall in the red area are roughly 68%.

+3+2+1+1-2-3

68.26%

random score

The probability of a random or chance event happening in any specific region of the curve is also equal to the percentage of the total area represented by that region

the odds of a chance event happening two standard deviations beyond the mean are approximately 4.28%, or less than 5%

The odds of a randomor chance event

happening in thisregion are 2.14%

The odds of a randomor chance event

happening in thisregion are 2.14%

+3+2+1+1-2-3

When a researcher states that his/her results are significant at the p < .05 level, the researcher means the results depart so much from what would be expected by chance that he/she is 95% confident they could not have been obtained by chance alone.

The results are probably due to the experimental manipulation, and not due to chance

By chance alone, results should wind up in either of these two regions less than 5% of the time

-1-2 +2+1-3 +3

When a researcher states that his/her results are significant at the p < .01 level, the researcher means the results depart so much from what would be expected by chance alone, that he/she is 99% confident they could not have been obtained merely by chance.

The results are probably due to the experimental manipulation and not to chance

By chance alone, results should wind up in either of these two regions less than 1% of the time.

-1-2 +2+1-3 +3

When a researcher employs a nondirectional hypothesis, the researcher is expecting a significant difference at either “tail” of the curve.

When a researcher employs a directional hypothesis, the researcher expects a significant difference at one specific “tail” of the curve.

-1-2 +2+1-3 +3

Nondirectional hypothesis

either tail of the curve

Directional hypothesis

one tail or the other

The “control” group in an experiment represents normalcy.

Scores for a “control” group are expected to be typical, or “average.”

The “treatment” group in an experiment is exposed to a manipulation or stimulus condition.

Scores for a “treatment” group are expected to be significantly different from those of the control group.

The researcher expects the “treatment” group to be 2 std. dev. beyond the mean of the control group.

-1-2 +2+1-3 +3

The control group should be in the middle of the distribution

The treatment group is expected to be 2 std. dev beyond the mean