Describing Distributions Means Standard deviation Z scores Normal distribution Norms Tracking

Means The arithmetic average score in a distribution is called the mean.

Standard Deviations You can obtain the average squared deviation around the mean, known as the variance. We need to take the square root of the variance. The square root of the variance is the standard deviation.

The standard deviation is thus the square root of the average squared deviation around the mean. Although the standard deviation is not an average deviation, it gives a useful approximation of how much a typical score is above or below the average score. Also, we divide by N - 1 rather than N to recognize that S of a sample is only an estimate of the variance of the population.

Z Score A Z score is the difference between a score and the mean, divided by the standard deviation. If a score is equal to the mean, then its Z score is O. For example, suppose the score and the mean are both 6; then = O. Zero divided by anything is still O. If the score is greater than the mean, then the Z score is positive. If the score is less than the mean, then the Z score is negative.

Normal distribution? Is it natural? Is it something intended by God? Yes, natural phenomena are naturally distributed. Heights, Weights and even human intelligence.

In normal distribution, extremely-large values and extremely-small values are rare and occur near the tail ends. Most-frequent values are clustered around the mean (which here is same as the median and mode) and fall off smoothly in either side of it.valuestailmedianmode In normal distribution, 68 percent of all values lie within one standard deviation, percent within two standard deviations, and 99.8 within three standard deviations (called six sigma in quality control).percentstandard deviationscalled six sigmaquality control In other words, only one out of a thousand values will fall outside of six sigma. This distribution is called 'normal' in the sense of 'ideal' or 'standard' against which other distributions may be compared.words distributions

Norms If you test all people in the population, then their scores will be distributed naturally in a bell shape. The norms for a test are based on the distribution of scores obtained by some defined sample of individuals. Norms are usually assumed as a normal distribution. Norms are used to give information about performance relative to what has been observed in a standardization sample.

McCall’s T = 10Z + 50 Qyartiles: divide the frequency distribution into equal fourths (25% intervals). Deciles divide: the frequency distribution into equal 10ths (10% intervals). Stanine system: ranges from 1 to 9 with a mean of 5 and a SD of approximately 2. IQ: a mean of 100 and SD of 15 used for most of intelligent tests

Tracking Children who are small as infants often remain small and continue to grow at a slower pace than do others. This tendency to stay at about the same level relative to one’s peers is known as tracking. Height and weight are good examples of physical characteristics that track. Figures

An example of a child going off track.

Questions My Z score is 0.8. Am I above average? Can you estimate my percentile? Crystal got 89 (Z score 1.1) in her class exam and Leo got 93( Z score.07) in his class exam. (They are in different classes) Who did better? If you test all the people in a population, their scores will be naturally distributed in a symmetrical bell shape. It is called ____________. If my IQ is 130, my percentile rank is _ and my Z score ______. Why do we need norms?