# A.k.a. “bell curve”.  If a characteristic is normally distributed in a population, the distribution of scores measuring that characteristic will form.

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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 scores medium scores higher scores

 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 mean median mode

 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%

 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% 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 -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 random or chance event happening in this region are 2.14% The odds of a random or chance event happening in this region are 2.14% +3+2+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 -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. -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. -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. -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

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