Standardizing Data and Normal Model(C6 BVD)

C6: Z-scores and Normal Model

*AP Statistics Review

Transforming Data

*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?

*Standardized value (Z-score)

*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.

*Density Curves

*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).

*Normal Distributions

*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).

*68-95-99.7 rule

*Example

*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.

*Finding Normal Percentiles

*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?

*1st quartile – 25th 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.

* Is it Normal?

*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 68-95-99.7 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 68-95-99.7 rule or z-table to find probabilities/areas/proportions under the density curve.