C6 Normal Distributions and Standard Scores

• Law of Frequency of Error or the Normal Curve, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. 
• Properties of the Normal Curve 1. normal curves are symmetrical: 2. they are unimodal: the mean, median, and mode all have the same value. 3. normal curves have that familiar bell-shaped form. 4. the tails never actually touch the horizontal axis.
• Properties of the Normal Curve: the equation of the normal curve describes a family of distributions.
• The proportion of area under any part of a frequency curve is equal to the proportion of cases in the same location.
• z score formula: z = (x - mean) / S
• A z-score states how many standard deviation units the original score lies above or below the mean of its distribution.
• A standard score expresses a score’s position in relation to the mean of the distribution, using the standard deviation as the unit of measurement.
• The normal distribution has wide applicability in both descriptive and inferential statistics.
• The z scores are also called standard scores because they have been standardized to a mean of 0 and a standard deviation of 1.
• Two fundamental problems involve the normal curve:finding area under the curve (proportion of cases) when the score location is known, and finding score locations when the area is known. The normal curve also can help interpret the difference between two means when this value is expressed as an effect size.
• Because z scores involve awkward decimals and negative values, standard scores of other kinds have been devised, such as T scores (which have a mean of 50 and a standard deviation of 10).
• Standard scores add meaning to raw scores because they provide a frame of reference by permitting comparison of scores from different distributions. It is important always to keep in mind the nature of the reference group from which these scores derive, because it affects interpretation.
• Percentile ranks, like standard scores, derive their meaning by comparing an individual performance with that of a known reference group.
• the“proportion above”a score is synonymous with the“area beyond”it.
• The empirical rule in statistics states that roughly 68% of data falls within 1 standard deviation of the mean, 95% falls within 2 standard deviations, and 99.7% falls within 3 standard deviations. This is characteristic of normal distributions only.
• In a normal distribution the bottom 75% of the cases fall below 0.67.
• From the standard normal table, we get, the value of z such that P ( 0 < Z < z ) = 0.4 is equal to 1.28. Thus, the z-score that divides the upper 20 cases from the bottom 180 cases for a normal distribution is 1.28.
• As per the z-score distribution table for a normal distribution, the most extreme 5% of all cases on both sides of the center correspond to a Z-score of 1.96.
• The middle 20% of a normal distribution is located between z = –0.25 and z = +0.25.
• o calculate a z-score, subtract the mean from the raw score and divide that answer by the standard deviation. (i.e., raw score =15, mean = 10, standard deviation = 4. Therefore 15 minus 10 equals 5. 5 divided by 4 equals 1.25.