SMHS CenterSpreadShape - Revision history

Glenbrau: /* Distribution Shape */

2015-03-27T18:59:53Z

‎Distribution Shape

Glenbrau: /* Problems */

2015-03-27T18:59:35Z

‎Problems

Glenbrau: /* Distribution Shape */

2015-03-27T18:57:50Z

‎Distribution Shape

Glenbrau: /* Applications */

2015-03-27T18:57:05Z

‎Applications

Glenbrau: /* Variance and Standard Deviation */

2015-03-27T18:53:53Z

‎Variance and Standard Deviation

Glenbrau: /* Five-number summary */

2015-03-27T18:51:19Z

‎Five-number summary

Glenbrau: /* Measures of Variation and Dispersion */

2015-03-27T18:49:47Z

‎Measures of Variation and Dispersion

Glenbrau: /* Other Centrality Measures */

2015-03-27T17:21:53Z

‎Other Centrality Measures

Glenbrau: /* Resistant Mean-related Measures of Centrality */

2015-03-27T17:19:30Z

‎Resistant Mean-related Measures of Centrality

Glenbrau: /* Median */

2015-03-27T17:17:24Z

‎Median

@@ Line 140: / Line 140: @@
 ====Distribution Shape====
 * A distribution is '''unimodal''' if it has one mode. Unimodal distributions include:
-** Bell shaped distributions (symmetric, normal)
+** Bell shaped distributions (symmetric, Normal)
 ** Skewed right or skewed left
 ** We can use the [[AP_Statistics_Curriculum_2007_EDA_Center | mean and median]] to help interpret the shape of a distribution. In general, for an unimodal distribution we have these properties:

@@ Line 216: / Line 216: @@
 * Collect data, draw the sample histogram or dot-plot and classify the shape of the distribution accordingly.  Also, if unimodal, classify symmetry (symmetric, skewed right or skewed left).
-** Data collected on height of randomly sampled college students.
+** Data collected on height of randomly sampled college students
-** Data collected on height of randomly sampled female college students.
+** Data collected on height of randomly sampled female college students
-** The salaries of all persons employed by a large university.
+** The salaries of all persons employed by a large university
-** The amount of time spent by students on a difficult exam.
+** The amount of time spent by students on a difficult exam
-** The grade distribution on a difficult exam.
+** The grade distribution on a difficult exam
 ===References===

@@ Line 143: / Line 143: @@
 ** Skewed right or skewed left
 ** We can use the [[AP_Statistics_Curriculum_2007_EDA_Center | mean and median]] to help interpret the shape of a distribution. In general, for an unimodal distribution we have these properties:
-*** If '''mean = median''', the distribution is symmetric
+*** If '''mean = median''', the distribution is symmetric.
-*** If '''mean > median''', the distribution is right skewed
+*** If '''mean > median''', the distribution is right skewed.
-*** If '''mean < median''', the distribution is left skewed
+*** If '''mean < median''', the distribution is left skewed.
 *** Note that depending on the protocol for computing the median, these rules may be violated for certain distributions. Some [http://www.amstat.org/publications/jse/v13n2/vonhippel.html examples of such violations are included in this JSE paper].

@@ Line 140: / Line 140: @@
 ====Distribution Shape====
 * A distribution is '''unimodal''' if it has one mode. Unimodal distributions include:
-** Bell shaped distributions (symmetric, Normal)
+** Bell shaped distributions (symmetric, normal)
 ** Skewed right or skewed left
 ** We can use the [[AP_Statistics_Curriculum_2007_EDA_Center | mean and median]] to help interpret the shape of a distribution. In general, for an unimodal distribution we have these properties:
-*** If '''mean = median''', then the distribution is symmetric
+*** If '''mean = median''', the distribution is symmetric
-*** If '''mean > median''', then the distribution is right skewed
+*** If '''mean > median''', the distribution is right skewed
-*** If '''mean < median''', then the distribution is left skewed
+*** If '''mean < median''', the distribution is left skewed
 *** Note that depending on the protocol for computing the median, these rules may be violated for certain distributions. Some [http://www.amstat.org/publications/jse/v13n2/vonhippel.html examples of such violations are included in this JSE paper].

@@ Line 102: / Line 102: @@
 =====Variance and Standard Deviation=====
-The logic behind the variance and standard deviation measures is to measure the difference between each observation and the mean (i.e., dispersion). Suppose we have ''n > 1'' observations, <math>\left \{ y_1, y_2, y_3, ..., y_n \right \}</math>. The deviation of the <math>i^{th}</math> measurement, <math>y_i</math>, from the mean (<math>\overline{y}</math>) is defined by <math>(y_i - \overline{y})</math>.
+The purpose of the variance and standard deviation measures is to measure the difference between each observation and the mean (i.e., dispersion). Suppose we have ''n > 1'' observations, <math>\left \{ y_1, y_2, y_3, ..., y_n \right \}</math>. The deviation of the <math>i^{th}</math> measurement, <math>y_i</math>, from the mean (<math>\overline{y}</math>) is defined by <math>(y_i - \overline{y})</math>.
 Does the average of these deviations seem like a reasonable way to find an average deviation for the sample or the population? No, because the sum of all deviations is trivial:

@@ Line 97: / Line 97: @@
 =====Five-number summary=====
-The five-number summary for a dataset is the 5-tuple <math>\{min, Q_1, Q_2, Q_3, max\}</math>, containing the sample minimum, first-quartile, second-quartile (median), third-quartile, and maximum.
+The five-number summary for a dataset is the 5-tuple <math>\{min, Q_1, Q_2, Q_3, max\}</math>, which contains the sample minimum, first-quartile, second-quartile (median), third-quartile, and maximum.
 For example, the 5-number summary for the [[SMHS_CenterSpreadShape#Median| Long Jump data above]] is $(60,68,76,80,106)$.

@@ Line 77: / Line 77: @@
 ====Measures of Variation and Dispersion====
-There are many measures of (population or sample) variation, e.g., the range, the variance, the standard deviation, mean absolute deviation, etc. These are used to assess the dispersion or spread of the population.
+There are many measures of (population or sample) variation, including the range, the variance, the standard deviation, mean absolute deviation, etc. These are used to assess the dispersion or spread of the population.
 =====Range=====
-The range is the easiest measure of dispersion to calculate, yet, perhaps not the best measure. The '''Range = max - min'''. For example, for the survival time data, the range is calculated by:
+The range is the easiest measure of dispersion to calculate, but is probably not the best measure. The '''Range = max - min'''. For example, for the survival time data, the range is calculated by:
-<center>Range = 106 – 60 = 46.</center> Note that the range is only sensitive to the extreme values of a sample and ignores all other information. So, two completely different distributions may have the same range.
+<center>Range = 106 – 60 = 46.</center> Note that the range is only sensitive to the extreme values of a sample and ignores all other information. Therefore, two completely different distributions may have the same range.
 =====Quartiles and IQR=====
-The first quartile (<math>Q_1</math>) and the third quartile (<math>Q_3</math>) are defined values that split the dataset into ''bottom-25% vs. top-75%'' and ''bottom-75% vs. top-25%'', respectively. Thus the inter-quartile range (IQR), which is the difference <math>Q_3 - Q_1</math>, represents the central 50% of the data and can be considered as a measure of data dispersion or variation. The wider the IQR, the more variant the data.
+The first quartile (<math>Q_1</math>) and the third quartile (<math>Q_3</math>) are defined values that split the dataset into ''bottom-25% vs. top-75%'' and ''bottom-75% vs. top-25%'', respectively. Thus, the inter-quartile range (IQR), which is the difference <math>Q_3 - Q_1</math>, represents the central 50% of the data and can be considered as a measure of data dispersion or variation. The wider the IQR, the more variant the data.
-For example, <math>Q_1=68</math>,  <math>Q_3=80</math> and <math>IQR=Q_3-Q_1=12</math>, for the survival time data shown above. Thus, we expect the middle half of all survival times (for that patient cohort and clinical intervention) to be between 68 and 80 days.
+For example, <math>Q_1=68</math>,  <math>Q_3=80</math> and <math>IQR=Q_3-Q_1=12</math> for the survival time data shown above. Thus, we expect the middle half of all survival times (for that patient cohort and clinical intervention) to be between 68 and 80 days.
 =====Coefficient of Variation=====
-For a given process, the coefficient of variation (CV) is defined as the ratio of the standard deviation (<math>\sigma </math>) to the mean (<math>\mu </math>):
+For a given process, the coefficient of variation ($CV$) is defined as the ratio of the standard deviation (<math>\sigma </math>) to the mean (<math>\mu </math>):
-:<math>CV = \frac{\sigma}{\mu}.</math>
+:<math>CV = \frac{\sigma}{\mu}</math>
-Obviously, the CV is well-defined for processes with well-defined first two moments (mean and variance), but also requires a ''non-trivial'' mean (<math>\mu \not= 0</math>). If the CV is expressed as percentage, than this ratio is  multiplied by 100. The sample coefficient of variation is computed mostly for data measured on a ratio scale. For instance, if a set of distances are measured, the standard deviation does not depend on whether the distances were measured in kilometers (metric) or miles. This is because changes in the particle/object's distances by 1 kilometer also changes its distance by 1 mile. However the mean distance of the data would differ in each measurement scale (as [http://en.wikipedia.org/wiki/Mile 1 mile is approximately 1.7 kilometers]) and thus the coefficient of variation would differ. In general, the CV may not have any meaning for data on an interval scale.
+Obviously, the $CV$ is well-defined for processes with well-defined first two moments (mean and variance), but also requires a ''non-trivial'' mean (<math>\mu \not= 0</math>). If the $CV$ is expressed as a percentage, this ratio is  multiplied by 100. The sample coefficient of variation is computed mostly for data measured on a ratio scale. For instance, if a set of distances are measured, the standard deviation does not depend on whether the distances were measured in kilometers (metric) or miles. This is because changes in the particle/object's distances by 1 kilometer also changes its distance by 1 mile. However the mean distance of the data would differ in each measurement scale (as [http://en.wikipedia.org/wiki/Mile 1 mile is approximately 1.7 kilometers]) and thus the coefficient of variation would differ. In general, the $CV$ may not have any meaning for data on an interval scale.
-The sample-coefficient of variation is computed by plugin the sample-driven estimates of the standard deviation (sample-standard deviation, <math>s</math>, and the sample-average, <math>\bar{x}</math>). In image processing, the reciprocal of the coefficient of variation is  μ/σ is called ''signal-to-noise-ratio'' (SNR).
+The sample-coefficient of variation is computed by plugin the sample-driven estimates of the standard deviation (sample-standard deviation, <math>s</math>, and the sample-average, <math>\bar{x}</math>). In image processing, the reciprocal of the coefficient of variation is  μ/σ is called ''signal-to-noise-ratio'' ($SNR$).
 =====Five-number summary=====
@@ Line 107: / Line 107: @@
 <center><math>\sum_{i=1}^n{(y_i - \overline{y})}=0.</math></center>
-To solve this problem we employ different versions of the '''mean absolute deviation''':
+To solve this problem, we employ different versions of the '''mean absolute deviation''':
 <center><math>{1 \over n-1}\sum_{i=1}^n{|y_i - \overline{y}|}.</math></center>

@@ Line 61: / Line 61: @@
 ====Other Centrality Measures====
-The ''arithmetic'' mean answers the question, ''if all observations were equal, what would that value (''center'') have to be in order to achieve the same <u>total</u>?''
+The ''arithmetic'' mean answers the question: ''If all observations were equal, what would that value ''(center)'' have to be in order to achieve the same <u>total</u>?''
 : <math>n\times \bar{x}=\sum_{i=1}^n{x_i}</math>
@@ Line 67: / Line 67: @@
 =====Harmonic Mean=====
-If we study speeds (velocities) the ''arithmetic'' mean is inappropriate. However the [http://en.wikibooks.org/wiki/Statistics/Summary/Averages/Harmonic_Mean harmonic mean (computed differently)] gives the most intuitive answer to what the "''middle''" is for a process. The harmonic mean answers the question ''if all the observations were equal, what would that value have to be in order to achieve the same sample <u>sum of reciprocals</u>?''
+If we study speeds (velocities), the ''arithmetic'' mean is inappropriate. However, the [http://en.wikibooks.org/wiki/Statistics/Summary/Averages/Harmonic_Mean harmonic mean (which is computed differently)] gives the most intuitive answer to what the "''middle''" is for a process. The harmonic mean answers the question: ''If all the observations were equal, what would that value have to be in order to achieve the same sample <u>sum of reciprocals</u>?''
 : ''Harmonic mean'': <math>\hat{\hat{x}}= \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \ldots + \frac{1}{x_n}}</math>
 =====Geometric Mean=====
-In contrast, the [http://en.wikibooks.org/wiki/Statistics/Summary/Averages/Geometric_Mean geometric mean] answers the question ''if all the observations were equal, what would that value have to be in order to achieve the same sample <u>product</u>?''
+In contrast, the [http://en.wikibooks.org/wiki/Statistics/Summary/Averages/Geometric_Mean geometric mean] answers the question: ''If all the observations were equal, what would that value have to be in order to achieve the same sample <u>product</u>?''
 : ''Geometric mean'': <math>\tilde{x}^n={\prod_{i=1}^n x_i}</math>
 : Alternatively: <math>\tilde{x}= \exp \left(  \frac{1}{n} \sum_{i=1}^n\log(x_i) \right)</math>

@@ Line 52: / Line 52: @@
 ====Resistant Mean-related Measures of Centrality====
-The following two sample measures of population centrality estimate resemble the calculations of the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Expectation_.28Mean.29 | mean]], however they are much more ''resistant'' to change in the presence of outliers.
+The following two sample measures of population centrality estimate resemble the calculations of the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Expectation_.28Mean.29 | mean]]; however, they are much more ''resistant'' to change in the presence of outliers.
 =====K-times trimmed mean=====
@@ Line 58: / Line 58: @@
 =====Winsorized k-times mean=====
 The Winsorized k-times mean is defined similarly by <math>\bar{y}_{w,k}={1\over n}( k\times y_{(k)}+\sum_{i=k+1}^{n-k}{y_{(i)}}+k\times y_{(n-k+1)})</math>, where <math>k\geq 0</math> is the trim-factor and <math>y_{(i)}</math> are the order statistics (small to large). In this case, before we compute the arithmetic average, we replace the ''k'' smallest and the ''k'' largest observations with the k<sup>th</sup> and (n-k)<sup>th</sup> largest observations, respectively.
 ====Other Centrality Measures====

@@ Line 28: / Line 28: @@
 =====Median=====
 The '''sample-median''' can be thought of as the point that divides a distribution in half (50/50). The following steps are used to find the sample-median:
-* Arrange the data in ascending order
+* Arrange the data in ascending order.
-* If the sample size is odd, the median is the middle value of the ordered collection
+* If the sample size is odd, the median is the middle value of the ordered collection.
 * If the sample size is even, the median is the average of the middle two values in the ordered collection.
 * For the survival time example data above, we have the following ordered data: