PROPERTIES OF MEAN, MEDIAN

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1 PROPERTIES OF MEAN, MEDIAN In the last class quantitative and numerical variables bar charts, histograms(in recitation) Mean, Median Suppose the data set is {30, 40, 60, 80, 90, 120} X = 70, median = 70 Suppose we add a large number 1000 to the list, then the new X = 202 median = 80 One extreme value changes the mean dramatically but the median does not change much 1 Median is Robust. A few outliers will have no or small effect on the median. Mean can change dramatically in the presence of a very large or very small value. 2 If x 1, x 2,..., x m have mean x and median m then ax 1 + b, ax 2 + b,..., ax n + b will have mean a x + b and median am + b. ( Mean respects change of scale) [.5cm] 3 Mean is mathematically more easier to work with than median. Mean or Median which is better? Depends on the context. If there are extreme values present in the data,then Median is better than Mean Mean is easier to compute. Easy to update if you have additional observations. Not so with median

2 MEASURES OF SPREAD data:x 1, x 2,...,..., x n (x 1 x), (x n x),..., (x n x) are deviations from the mean x n n n (x i x) = x i x 1 1 = n x n x = 0 The total of deviations above the mean is same as the total of deviations below the mean contrast with median: the number of observations above median same as number of observations below median 1 1. Range: Maximum - Minimum 2. IQR (Inter Quartile Range)= Q 3 Q 1 (Minimum,Q 1, Median, Q 3, Maximum: Five number Summary) 3. Standard Deviation PERCENTILES STANDARD DEVIATION p-th percentile: The value below which (roughly) p% of the data points lie. When p=50 we get the median. The 25-th percentile is called FIRST QUARTILE and denoted by Q 1 The 75-th percentile is called THIRD QUARTILE and denoted by Q 3 Standard deviation is another, very useful, measure of the amount of deviation from the mean. These are useful in determining Outliers and can be represented in a Box plot.

3 STANDARD DEVIATION Computation STANDARD DEVIATION Computation Let x 1, x 2,..., x n be a data set with mean x, x-values (x x) (x x) 2 x 1 (x 1 x) (x 1 x) 2 x 2 (x 2 x) (x 2 x) 2 x 3 (x 3 x) (x 3 x) x n (x n x) 0 (x n x) 2 n 1 (x i x 2 Population variance : σ 2 n i=1 = (x i x) 2 n Population standard deviation: σ = Population Variance Sample variance: s 2 n i=1 = (x i x) 2 n 1 Sample standard deviation: s = Sample Variance Why the square in (x x) 2? (xi x) = x i n x = 0. Some of the deviations from the mean are positive and some negative. When you sum they cancel out each other. So take square to make everything positive Why take the square root of the variance? This ensures that the measure of variability standard deviation is in the same unit as the data.

4 Example The following data are number of passengers on flights of Delta Air Lines between San Francisco and Seattle over 33 days in April and early May. 128,121,134,136,136,118,123,109,120,116,125,128,121,129,130,131, 127,119,114,134,110,136,134,125,128,123,128,133,132,136,134,129, 132 Find the range, variance and standard deviation of the data (assumed to be a sample) Maximum is 136, Minimum is 109 So the range is = 27 X = Properties of Variance 1. x-values (x x) (x x) ( ) ( ) 2 = ( ) ( ) 2 = ( ) ( ) 2 = ( ) 2 = Sample variance = /32 = Sample s.d = = 7.6 Var (x 1, x 2,..., x n) 0 and is = 0 only when x 1 = x 2 =... = x n Var (x 1 + b, x 2 + b,..., x n + b) = Var(x 1, x 2,..., x n) Adding a constant does not change the variance Var (ax 1, ax 2,..., ax n) = a 2 Var(x 1, x 2,..., x n) Var(ax 1 + b, ax 2 + b,..., ax n + b) = a 2 Var(x 1, x 2,..., x n)

5 Properties of Standard Deviation HISTOGRAM Histogram of x1 Histogram of x2 S.D (x 1, x 2,..., x n) 0 and is = 0 only when x 1 = x 2 =... = x n S.D (x 1 + b, x 2 + b,..., x n + b) = S.D(x 1, x 2,..., x n) Density x1 Density x2 Adding a constant does not change the standard deviation S.D (ax 1, ax 2,..., ax n) = a S.D(x 1, x 2,..., x n) S.D (ax 1 + b, ax 2 + b,..., ax n + b) = a S.D(x 1, x 2,..., x n) Density 0.00 Histogram of x Density 0.0 Histogram of x x3 x5 Figure: Histograms What to look for in a Histogram PROPERTIES OF MEAN, MEDIAN Shape 1. Is it symmetric? skewed? 2. Does it have one mode? two modes? three modes? [ Peaks are called modes] Two modes suggests that the there are two subgroups present 3. Outliers? (will return to this later) 4. Gaps? 1. Describes how data are distributed 2. Measures of Shape Skew = Symmetry Left-Skewed Mean Median Symmetric Mean = Median Right-Skewed Median Mean

6 Transformation to get to symmetry If the data is nearly symmetric, mean is a good measure of center For skewed data, median is a better measure Generally most statistical method use mean and symmetric bell shaped distn How do we justify this? NOT INCLUDED IN EXAM Sometimes you can transform the data and get a symmetric distn Transformation to get to symmetry Transformation to get to symmetry transformation.pdf Frequency Histogram of y y Frequency histogram of log y x

7 histogram2.pdf histogram of x by square.pdf histogram of x^2 Frequency Frequency y x What to look for in a Histogram Bimodal histogram 1. Is it symmetric? skewed? 2. Does it have one mode? two modes? three modes? [ Peaks are called modes] Two modes suggests that the there are two subgroups present 3. Outliers? (will return to this later) 4. Gaps? 0.6 * dnorm(x, 1, 1) * dnorm(x, 5, 1) Index

8 outliers outliers are data points that are far away from the main body of the histogram outliers need to be investigated further. Typically outliers can be explained. by education.png Figure: scatter plot age vs wage Figure: Histogram by education

9 my height is 5 7" It is a reasonable height in the general population In the population of basket ball players, this would be an outlier comparison.png Figure: outliers CHEBYSHEV s RULE Outliers are data points far away from the center of the data distribution far away depends on the center and on the amount of variation in the data mean and s.d determine which data points are outliers Atleast (1 1 k 2 ) part of the histogram lies within ks of the mean k=2: At least 75 % of the observations lie within 2 standard deviations of the mean k=3: at least 8/9, approx 90% of the observations lie within 3 standard deviations of the mean

10 If the histogram is bell shaped then prob 153, 144 Approximately 68 % of the observations lie within x s, x + s Approximately 95 % of the observations lie within x 2s, x + 2s Approximately 99.7 % of the observations lie within x 3s, x + 3s Problem If the range of a set of data is 20, find a rough approximation to the s.d of the data set 75% of the data falls within x 2s, x + 2s i.e within a range of x + 2s x 2s = 4s so 4s range = 20 so s 20 4 = 5 Numerical measures of relative standing Let µ be the mean of a data set when the data set is the population σ be the s.d of a data set when the data set is the population x be the mean of a data set when the data set is a sample σ be the sd of a data set when the data set is a sample For any value x, The Population z-score of x is The Sample z-score of x is z = x µ σ z = x X s The z- score is a measure of how many s.d s is x away from the mean

11 x is k standard deviations away from the mean is same as x µ > kσ. i.e. If x µ > kσ then x µ σ > k i.e z > k. So x is larger than µ + kσ is equivalent to z-value of x is larger than k. Similarly, x is smaller than µ kσ is equivalent to z-value of x is smaller than k. So in terms of z - values At least 75 % of the observations have z-values less than 2 at least 8/9, approx 90% of the observations have z-values less than 3 Put differently, At most 10% of the observations have z-values larger than 3 If the histogram is bell shaped then Since values that are far away from the mean have very large or very small (negative) z -scores, we can use z-scores to define outliers. Approximately 68 % of the observations have z-values less than 1 Observations with z-scores greater than 3 in absolute value are considered outliers. Approximately 95 % of the observations have z-values less than 2 Approximately 99.7 % of the observations have z-values less than 3

12 problems 139,140,161 Another way to define outliers is via interquartile range and box plots. Outliers 2 Minimum, Q 1, Median, Q 3, Maximum Plot1.pdf are called Five number Summary Q 2 Q 1 is called Inter Quartile Range [IQR] 1. Graphical display of data using 5-number summary data below Q 1 1.5(IQR) and data aboveq (IQR) are called Outliers X smallest Q 1 Median Q 3 X largest These are displayed in a

13 Plot2.pdf Plot3.pdf 1. Draw a rectangle (box) with the ends (hinges) drawn at the lower and upper quartiles (Q L and Q U ). The median data is shown by a line or symbol (such as + ). 2. The points at distances 1.5(IQR) from each hinge define the inner fences of the data set. Line (whiskers) are drawn from each hinge to the most extreme measurements inside the inner fence. 3. The symbol (*) represents measurements falling beyond the inner fences. 4. Symbols that represent the median and extreme data points vary depending on software used. You may use your own symbols if you are constructing a box plot by hand. Plot4.pdf Shape & Plot5.pdf Detecting Outliers Left-Skewed Symmetric Right-Skewed Q 1 Median Q 3 Q 1 Median Q 3 Q 1 Median Q 3 s: Observations falling between the inner and outer fences are deemed suspect outliers. Observations falling beyond the outer fence are deemed highly suspect outliers. z-scores: Observations with z-scores greater than 3 in absolute value are considered outliers. (For some highly skewed data sets, observations with z-scores greater than 2 in absolute value may be outliers.)

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