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Standardizing Data and Normal Model(C6 BVD) C6: Z-scores and Normal Model

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* Imagine a list of data, such as (1,3,5,7,9). * If you add/subtract something to all the data, what happens to center (mean)? Spread (Sx)? * If you multiply/divide all the data by something, what happens to center? Spread? * If you subtracted the mean from all the data, what would the mean of the transformed list be? * If you divided all data in that list by Sx, what would the new standard deviation be?

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* When you transform the data by subtracting the mean and dividing by Sx, the new list of data has a mean of 0 and a standard deviation of 1. You can do this to any data, no matter the shape of the distribution, units, etc. * If we then use the standard deviation as a “yard stick” to see how extraordinary a particular value is, we can compare values from any data sets, no matter how different the original distributions were. We can compare 100m dash times with discus tosses, etc. * Z = (value – mean)/Sx * A z-score tells you how many standard deviations above/below the mean a result is. The farther away it is from the mean, the more extraordinary or unusual it is.

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* Sometimes the overall pattern of a large number of observations is so regular we can describe it by a smooth curve, called a density curve. * The area under a density curve is always 1. * The area under the curve between any two intervals is the proportion of all observations that fall in that interval. * Median – divides curve into equal areas. * Mean – the balance (see-saw) point. * Median = Mean if the curve is symmetric. If it isn’t, mean is pulled in the direction of skew (the long tail).

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* Normal curves are a very useful class of density curves. They are symmetric, unimodal, bell- shaped. They are described by N(mean, standard deviation) –these are parameters, not statistics * The points of inflection are one standard deviation to either side of the mean. * There are an infinite number of normal curves. Your z-table is for the STANDARD NORMAL CURVE which has been transformed to a mean of zero and standard deviation of 1 (i.e. standardized to use with z-scores). * rule

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* The distribution of heights for U.S. women can be modeled by N(64.5,2.5) * What % have heights over 67? * Between 62 and 72 inches? * What if z-score is somewhere between the standard deviations? – Use z-table or calculator -- Distributions menu – normalcdf(lower bound, upper bound) * Less than 5 feet? Z = -1.8 * Remember: area in table is LEFT-side area.

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* Example: Blood cholesterol level in mg/dl of teens boys can be described by N(170,30). What is the first quartile of the distribution? * 1 st quartile – 25 th percentile. * Find.2500 or closest in z-table – read z. * Calculator – use invnorm(.25) – must write percentile as decimal. * Use z-score equation z = (x-170)/30 to solve for x.

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* Not every density curve that looks normal really is normal. Never say something is “normal” if is really is only approximately normal or just unimodal/symmetric. * How to check: * 1. Plot data in a dotplot, stemplot or histogram. Is data unimodal, symmetric, bell-shaped? * 2. Does the rule work? – Find mean and standard deviation. Are about 68% of data points within 1 Sx of mean? (etc.) * 3. Can use Normal Probability Plot on TI-calculator – look for straight diagonal line. * 4. If data are not approximately normal, you can still find z-scores, but you cannot use rule or z- table to find probabilities/areas/proportions under the density curve.

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