1/3/2017. Measures of Location. Summary Statistics: Measures of Location and Spread. Statistics versus Parameters. Measures of Spread

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1 Summary Statistics: Measures of Location and Spread Measures of Location Illustrate where the majority of locations are found: e.g., means, medians, modes Illustrate how variable the data are: e.g., standard deviation, variance, standard error Statistics versus Parameters Statistics describe the sample Parameters describe the [unknown?] population Measures of Location: Mean Arithmetic Mean All observations weighted equally in calculation Arithmetic Mean Unbaised estimate of if: Observations from random individuals Samples are independent of eachother Observations drawn from a large population that can be described by a normal random variable X~N(, ) 1

2 Bias and Sample Size Other Means Geometric Mean Example from exponential population growth: when numbers are multiplied on an arithmetic scale then can be added on a logarithmic scale... So it depends how you use the mean Other Means Median and Mode Harmonic Mean Uses the reciprocal (1/Y) of the data Sensitive to extreme values that are small Often in conservation biology: effective population size Median and Mode Median: the middle observation (unless tied) Mode: the observations that occurs most frequently Which measure of location? Arithmetic mean most common Supported by Central Limit Theorem Geometric mean most appropriate for multiplicative measures Median or Mode when distribution doesn t match a standard probability distribution 2

3 Which measure of location? Researchers use different measure of location to support different points of view... Pay attention to what measure is supplied and always be suspicious of any measure of location that is not accompanied by a measure of spread! Variance and Standard Deviation Sum of Squares (SS): Degrees of Freedom Variance The number of independent observations that we have for estimating statistical parameters For now... n-1 Unbiased estimate of 2 Standard Error of the Mean Variance Standard Deviation Think of the standard error (or the mean) as an estimate of the standard deviation of the POPULATION MEAN 3

4 Standard Error of the Mean Standard Error of the Mean If inference is about the sample: provide SD (s) If the inference is about the means: provide the SE Skewness, Kurtosis, and Central Moments A central moment is the average of the deviations of all observations in a dataset from the mean of the observations, raised to a power r: Skewness, Kurtosis, and Central Moments r = 1 (1 st moment) always 0 r = 2 (2 nd moment) is the variance Skewness r = 3 (3 rd moment) divided by s 3 = skewness Skewness describes how the sample differs in shape from a symmetrical distribution Skewness g 1 = 0 normal distribution g 1 > 0 right-skewed (longer tail of observations to the right of the mean g 1 < 0 left-skewed (longer tail of observations to the left of the mean 4

5 Skewness Kurtosis Based on 4 th central moment (r=4) Measures the extent to which the distribution is distributed in the tails versus the center of the distribution Kurtosis Kurtosis Clumped, or platykurtic distributions have g 2 < 0 (less probability in the tails) Leptokurtic distributions have g 2 > 0 (less probability in the center) Skewness and Kurtosis Should be tested, but both measures are sensitive to outliers... Box plots of quantiles can portray the distribution of data more accurately than plots of means and standard deviations Quantiles 5

6 Other Measures Coefficient of Variation (CV) Variability independent of the mean Coefficient of Dispersion For discrete variables varianceto-mean ratio Measure of clumping, but dependent on scale Distribution of Points For normally distributed random variables: 67% of observations occur within 1 SD of the mean 96% of observations occur within 2 SD of the mean 6

7 Interpretation: 95% of the time such an interval will contain the true value of NOT: there is a 95% chance that the true occurs within the interval it either does or does not... 7