Chapter 5 A Normal World
GED111/CDS111 Statistics in Modern Society What is Normal? The Normal Distribution Single peaked Symmetric Same point for mean, median, mode Bell/mourn shape Variation can be characterized by the standard deviation GED111/CDS111 Statistics in Modern Society
The Normal Distribution and Relative Frequencies The relative frequency for any range of data values is the area under the curve covering that range of values The total relative frequency for any data set must be 1, thus the total area under the normal distribution curve must equal 1 GED111/CDS111 Statistics in Modern Society
When Can We Expect a Normal Distribution? Most data values are clustered near the mean, giving the distribution a well-defined single peak Data values are spread evenly around the mean, making the distribution symmetric Larger deviations from the mean become increasingly rare, producing the tapering tails of the distribution Individual data values result from a combination of many different factors, such as genetic and environmental factors GED111/CDS111 Statistics in Modern Society
GED111/CDS111 Statistics in Modern Society Exercises Q11-18 p202 GED111/CDS111 Statistics in Modern Society
Properties of the Normal Distribution About 68% of the data points fall within 1 standard deviation of the mean About 95% of the data points fall within 2 standard deviations of the mean About 99.7% of the data points fall within 3 standard deviations of the mean GED111/CDS111 Statistics in Modern Society
GED111/CDS111 Statistics in Modern Society Applying the 68-95-99.7 Rule Guideline can be set to distinguish “unusual” values GED111/CDS111 Statistics in Modern Society
GED111/CDS111 Statistics in Modern Society Standard Scores The number of standard deviations a data value lies above or below the mean GED111/CDS111 Statistics in Modern Society
Standard Scores and Percentiles Standard scores can be converted to percentiles Table 5.1 p211 Percentage, relative frequency, proportion, probability GED111/CDS111 Statistics in Modern Society
The Central Limit Theorem Suppose we take many random samples of size n for a variable with any distribution (not necessarily a normal distribution) and record the distribution of the means of each sample The distribution of the means will be approximately a normal distribution for large sample sizes The mean of the distribution of means approaches the population mean, µ, for large samples sizes The standard deviation of the distribution of means approaches σ/√n for large sample sizes, where σ is the standard deviation of the population GED111/CDS111 Statistics in Modern Society
The Central Limit Theorem (cont.) Figure 5.26 p218 Software http://onlinestatbook.com/stat_sim/sampling_dist/index.html GED111/CDS111 Statistics in Modern Society
GED111/CDS111 Statistics in Modern Society Focus on Psychology Are We Smarter Than Our Parents? p229 GED111/CDS111 Statistics in Modern Society