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Statistical Techniques I EXST7005 Start here Measures of Dispersion

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n Objective - Hypothesis testing Background è We will test primarily means, but also variances - è Testing means requires a measure of the variability in the data set Course Progression

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MEASURES OF DISPERSION n These are measures of variation or variability among the elements (observations) of a data set n RANGE - difference between the largest and smallest observation è This is a rough estimator which does not use all of the information in the data set.

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MEASURES OF DISPERSION (continued) n Interquartile range - Q1 to Q3 (25% to 75%) better than range è What are Quartiles? – The first quartile (Q1) is the value that has one quarter of the values below (smaller than Q1) it and three quarters above it (larger than Q1) – The second quartile has half the values smaller and half the values larger – The third quartile has 3/4 smaller and 1/4 larger

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MEASURES OF DISPERSION (continued) è Percentile - a given percentile has that percent of the values below it and the remaining values above it. – e.g. The 40th percentile has 40% of the values smaller and 60% of the values larger n

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MEASURES OF DISPERSION (continued) n VARIANCE the "average" squared deviation from the mean. the POPULATION VARIANCE (called sigma squared) 2 è This is a parameter, and therefore a constant è where N is the size of the population

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n S2 is the SAMPLE VARIANCE (called s- squared). è This is a statistic, and therefore a variable è where n is the size of the sample NOTE that the divisor is n-1 rather than n. If n is used then the calculation is a biased estimator of 2. MEASURES OF DISPERSION (continued)

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n STANDARD DEVIATION a standard measure of the deviation of observations from the mean. è it is calculated as the square root of the variance = 2 this is a parameter S = S2 this is a statistic

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è the VARIANCE is the average squared deviation, so we take the square root of this to get back to the same units. è Absolute Mean Deviation the "average deviation" from the mean, but using absolute values. This is another possibility, but is not used much because the Variance is more flexible. MEASURES OF DISPERSION (continued)

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n A valid, useful measure of dispersion should è use all of the available information è be independent of other parameters (and statistics) for large data sets è be capable of being expressed in the same units as the variables è be small when the spread among the points in the data set is small, and large when the spread is wider. n The Standard deviation fills these criteria. MEASURES OF DISPERSION (continued)

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A note on UNITS n When we calculate the mean for a sample or population, the units on the mean are the same as for the original variable. è e.g. If the original variable was measured in inches, the units of the mean will be inches

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A note on UNITS (continued) n The variance also has units, but since the calculation involves the square of the original variable, the units on the variance are the original variable squared è e.g. If the original variable was measured in inches, the units of the variance would be inches squared è However, since the standard deviation is the square root of the variance, it's units would again be the same as the original variable.

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Calculating the Variance n The variance can in many cases be calculated more easily with the "calculator formula". n

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Calculating the Variance (continued) n When we refer to sum of squares or SS, we will refer to the Corrected Sum of Squares, unless otherwise stated. I will generally denote uncorrected sums of squares as UCSS or USS. n The calculator formula is then calculated as

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An example of variance n The CORRECTION FACTOR is an adjustment for the MEAN. è Examine two samples; Sample 1: 1, 2, 3 Y = 2 Sample 2: 11, 12, 13 Y = 12 n Note that the deviations from the mean are the same in each case è (-1, 0, 1). n and that SS = (-12)+(0)2+(1)2=2 for both samples

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An example of variance (continued) n The SS are è Sample 1 SS = SS = 2 è Sample 2 SS = SS = 2 n And the Variance for both samples is then SS / (n-1) = 2 / 2 = 1 n So two different looking sets of numbers have the same "scatter" and the same variance.

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Degrees of Freedom n Note that in the formula for a Population the divisor is N, while in the calculation for a sample the divisor is n-1 This occurs when the calculated estimate of one parameter (s2) uses an estimate of another parameter obtained from the same sample ( Y). Since we must use an estimate of to calculate our estimate of 2, the divisor is n-1; è this is called its degrees of freedom. If we needed to estimate two parameters prior to being able to estimate a parameter its degrees of freedom would be n-2.

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Degrees of Freedom (continued) Why? If we knew , then we could get a deviation from any single observation. – e.g. If we knew that =5, and we drew an observation at random and its value was 3, then the deviation would be -2. However, we cannot get an estimate of 2 from a single sample observation since that observation is also its own mean and the deviation is zero. – e.g. If we drew a single sample observation, with a value of 3, and we did not know the value of , then we would estimate the value of Y from our sample. That value would also be 3.

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Also, with more observations we can have each one deviating independently from , and the sum of the deviations has no restrictions. However, deviations from Y always sum to ZERO, so only the first n-1 can be "any" value, as soon as we know n-1 values, the last one is fixed by our knowledge of Y. Degrees of Freedom (continued)

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COEFFICIENT OF VARIATION n CV is the standard deviation expressed as a percent of the mean, e.g. CV = S / Y * 100% n the CV is used to compare relative variation between different experiments or different variables independent of the mean. n EXAMPLES: è compare the variability of peoples weights to peoples heights. è compare variation in infants lengths to adult heights.

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COEFFICIENT OF VARIATION (continued) n NUMERICAL EXAMPLE: compare the relative variation in fork length of fish to the weights and scale lengths of the same fish. Data from 3 year old Flier Sunfish (Centrarchus macropterus). Length (mm) Weight (g) Scale Lt. (mm) Mean Std Dev

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COEFFICIENT OF VARIATION (continued) n CV (length)= 15.1/131.8*100%=11.5% n CV (weight)=19.6/53.0*100%=37.0% n CV (scale length)=0.8/6.9*100%=11.6% n This calculation allow the comparison of different variables or variables on different scales. n Note: è the CV has no units è highly variable data may pass 100%

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SAS example (#1 continued) PROC UNIVARIATE DATA=ONE PLOT; VAR SALEPRIC; TITLE3 'Frequency table of house Sale Price'; RUN; Analysis of house sale price data Table 1.1 from Freund & Wilson, 1997 Frequency table of house Sale Price Univariate Procedure Variable=SALEPRIC Moments N 42 Sum Wgts 42 Mean Sum Std Dev Variance Skewness Kurtosis USS CSS CV Std Mean T:Mean= Pr>|T| Num ^= 0 42 Num > 0 42 M(Sign) 21 Pr>=|M| Sgn Rank Pr>=|S|

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SAS example (#1 continued) Quantiles(Def=5) 100% Max 75 99% 75 75% Q % % Med % % Q % 22 0% Min 15 5% 19 1% 15 Range 60 Q3-Q Mode 37 Extremes Lowest Obs Highest Obs 15( 3) 55.5( 35) 18.9( 4) 56.35( 39) 19( 1) 58.5( 40) 19.8( 2) 61.35( 41) 22( 13) 75( 42) PROC UNIVARIATE DATA=ONE PLOT; VAR SALEPRIC; TITLE3 'Frequency table of house Sale Price'; RUN;

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SAS example (#1 continued) Stem Leaf # Boxplot | | | *--+--* | | | Multiply Stem.Leaf by 10**+1 PROC UNIVARIATE DATA=ONE PLOT; VAR SALEPRIC; TITLE3 'Frequency table of house Sale Price'; RUN;

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SAS example (#1 continued) Analysis of house sale price data Table 1.1 from Freund & Wilson, 1997 Frequency table of house Sale Price Univariate Procedure Variable=SALEPRIC Normal Probability Plot * | +++ | ++++ | +++* | +*+** | +++** ****** | +**** | ***+*** | **++ | ++** | ++++ * *+++* ** PROC UNIVARIATE DATA=ONE PLOT; VAR SALEPRIC; TITLE3 'Frequency table of house Sale Price'; RUN;

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EXPECTED VALUES and BIAS n DEFINE n Unbiased Estimator: a statistic is said to be an unbiased estimator of a parameter if, with repeated sampling, the average of all sample statistics approaches the parameter. n Expected value: the mean value of a statistic from and infinitely large number of samples (the "long run" average).

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EXPECTED VALUES and BIAS (continued) n Note that in dividing by n-1 to calculate variance for a sample, results in a value which is LARGER than if we divide by n. If dividing by n-1 is the correct approach, it suggests that dividing by n causes a negative bias (a value which is, on the average, too SMALL). This is true.

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It is true that the expected value of the sample mean is equal to the population parameter. i.e. E{ Y} = , and it is also true that E{S2} = 2, so these are unbiased estimators. Note that for symmetric distributions, can also be estimated by the median, mode or midrange. However, the MEAN is an unbiased estimator of for all distributions. EXPECTED VALUES and BIAS (continued)

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n Expected Values are actually calculated as the sum (or integration for continuous variables) of the product of the observed values (Yi) in the distribution and the probability of occurrence of each value (p(Yi)). These have various uses, including the evaluation of bias.

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EXPECTED VALUES and BIAS (continued) n For our purposes; è The expected value is the measure of the true central tendency for the probability distribution. If we took all possible samples, the mean of them would be the expected value, provided the estimator we used is unbiased. è For any statistic, if the expected value of the statistic is the same as the population value, the statistic is unbiased.

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Summary of Dispersion n Dispersion is a measure of the variability among the elements of a population or sample n A number of estimates are available, including Range, Interquartile range, Variance and Standard deviation. All are available from SAS PROC UNIVARIATE. n Units of the variable are squared on variances, but the same as the original variable for standard deviations.

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Summary of Dispersion (continued) n Calculations on samples must consider degrees of freedom. n Both the sample mean and sample variance (when divided by "n-1") are unbiased estimators of the population mean and population variance.

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