b a c kn e x t h o m e Parameters and Statistics statistic A statistic is a descriptive measure computed from a sample of data. parameter A parameter is a descriptive measure computed from an entire population of data.

b a c kn e x t h o m e Measures of Central Tendency - Arithmetic Mean - arithmetic mean The arithmetic mean of a set of data is the sum of the data values divided by the number of observations.

b a c kn e x t h o m e Sample Mean If the data set is from a sample, then the sample mean,, is:

b a c kn e x t h o m e Population Mean If the data set is from a population, then the population mean, , is:

b a c kn e x t h o m e Measures of Central Tendency - Median - ordered array median An ordered array is an arrangement of data in either ascending or descending order. Once the data are arranged in ascending order, the median is the value such that 50% of the observations are smaller and 50% of the observations are larger.

b a c kn e x t h o m e Measures of Central Tendency - Median - median median 0.50(n+1)th ordered position If the sample size n is an odd number, the median, X m, is the middle observation. If the sample size n is an even number, the median, X m, is the average of the two middle observations. The median will be located in the 0.50(n+1)th ordered position.

b a c kn e x t h o m e Measures of Central Tendency - Mode - mode, The mode, if one exists, is the most frequently occurring observation in the sample or population.

b a c kn e x t h o m e Shape of the Distribution symmetric The shape of the distribution is said to be symmetric if the observations are balanced, or evenly distributed, about the mean. In a symmetric distribution the mean and median are equal.

b a c kn e x t h o m e Shape of the Distribution skewed positively skewed negatively skewed A distribution is skewed if the observations are not symmetrically distributed above and below the mean. A positively skewed (or skewed to the right) distribution has a tail that extends to the right in the direction of positive values. A negatively skewed (or skewed to the left) distribution has a tail that extends to the left in the direction of negative values.

b a c kn e x t h o m e Shapes of the Distribution

b a c kn e x t h o m e Measures of Variability - The Range - range The range is in a set of data is the difference between the largest and smallest observations

b a c kn e x t h o m e Measures of Variability - Sample Variance - sample variance, s 2, The sample variance, s 2, is the sum of the squared differences between each observation and the sample mean divided by the sample size minus 1.

b a c kn e x t h o m e s 2 Measures of Variability - Short-cut Formulas for s 2 sample variance, s 2, Short-cut formulas for the sample variance, s 2, are:

b a c kn e x t h o m e Measures of Variability - Population Variance - population variance,  2, The population variance,  2, is the sum of the squared differences between each observation and the population mean divided by the population size, N.

b a c kn e x t h o m e Measures of Variability - Sample Standard Deviation - sample standard deviation, s, The sample standard deviation, s, is the positive square root of the variance, and is defined as:

b a c kn e x t h o m e Measures of Variability - Population Standard Deviation- population standard deviation, , The population standard deviation, , is

b a c kn e x t h o m e The Empirical Rule (the 68%, 95%, or almost all rule) Empirical Rule For a set of data with a mound-shaped histogram, the Empirical Rule is: 68% approximately 68% of the observations are contained with a distance of one standard deviation around the mean;  1  95% approximately 95% of the observations are contained with a distance of 2 standard deviations around the mean;  2  almost all of the observations are contained with a distance of three standard deviation around the mean;  3 

b a c kn e x t h o m e Coefficient of Variation Coefficient of Variation, CV, The Coefficient of Variation, CV, is a measure of relative dispersion that expresses the standard deviation as a percentage of the mean (provided the mean is positive). sample coefficient of variation The sample coefficient of variation is

b a c kn e x t h o m e Coefficient of Variation population coefficient of variation The population coefficient of variation is

b a c kn e x t h o m e Percentiles and Quartiles Percentiles P th percentile Data must first be in ascending order. Percentiles separate large ordered data sets into 100ths. The P th percentile is a number such that P percent of all the observations are at or below that number. Quartiles Quartiles are descriptive measures that separate large ordered data sets into four quarters.

b a c kn e x t h o m e Percentiles and Quartiles first quartile, Q 1 25 th percentile The first quartile, Q 1, is another name for the 25 th percentile. The first quartile divides the ordered data such that 25% of the observations are at or below this value. Q 1 is located in the.25(n+1) st position when the data is in ascending order. That is,

b a c kn e x t h o m e Percentiles and Quartiles third quartile, Q 3 75 th percentile The third quartile, Q 3, is another name for the 75 th percentile. The first quartile divides the ordered data such that 75% of the observations are at or below this value. Q 3 is located in the.75(n+1)st position when the data is in ascending order. That is,

b a c kn e x t h o m e Interquartile Range Interquartile Range (IQR) The Interquartile Range (IQR) measures the spread in the middle 50% of the data; that is the difference between the observations at the 25 th and the 75 th percentiles: