Measures of Central Tendency: Ungrouped Data. Mode. Median. Mode -- Example. Median: Example with an Odd Number of Terms

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1 Measures of Central Tendency: Ungrouped Data Measures of central tendency yield information about particular places or locations in a group of numbers. Common Measures of Location Mode Median Percentiles Quartiles Mode The most frequently occurring value in a data set Applicable to all levels of data measurement (nominal, ordinal, interval, and ratio) Bimodal -- Data sets that have two modes Multimodal -- Data sets that contain more than two modes Stat 13D, UCLA, Ivo Dinov. 1 Stat 13D, UCLA, Ivo Dinov. Mode -- Example Median The mode is. There are more s than any other value Middle value in an ordered array of numbers. Applicable for ordinal, interval, and ratio data ot applicable for nominal data Unaffected by extremely large and extremely small values Stat 13D, UCLA, Ivo Dinov. 3 Stat 13D, UCLA, Ivo Dinov. 4 Median: Computational Procedure First Procedure Arrange the observations in an ordered array. If there is an odd number of terms, the median is the middle term of the ordered array. If there is an even number of terms, the median is the average of the middle two terms. Second Procedure The median s position in an ordered array is given by (n+1)/. Median: Example with an Odd umber of Terms Ordered Array There are 17 terms in the ordered array. Position of median (n+1)/ (17+1)/ 9 The median is the 9th term, 1. If the is replaced by 1, the median is 1. If the 3 is replaced by -13, the median is 1. Stat 13D, UCLA, Ivo Dinov. Stat 13D, UCLA, Ivo Dinov. 6 Page 1

2 Median: Example with an Even umber of Terms Ordered Array There are terms in the ordered array. Position of median (n+1)/ (+1)/ 8. The median is between the 8th and 9th terms, 14.. If the 1 is replaced by 1, the median is 14.. If the 3 is replaced by -88, the median is 14.. Arithmetic Commonly called the mean is the average of a group of numbers Applicable for interval and ratio data ot applicable for nominal or ordinal data Affected by each value in the data set, including extreme values Computed by summing all values in the data set and dividing the sum by the number of values in the data set Stat 13D, UCLA, Ivo Dinov. 7 Stat 13D, UCLA, Ivo Dinov. 8 Population X µ X X X X Sample 1 3 X X X Xn X X n n Stat 13D, UCLA, Ivo Dinov. 9 Stat 13D, UCLA, Ivo Dinov. 1 Percentiles Measures of central tendency that divide a group of data into 1 parts At least n% of the data lie below the nth percentile, and at most (1 - n)% of the data lie above the n th percentile Example: 9th percentile indicates that at least 9% of the data lie below it, and at most 1% of the data lie above it The median and the th percentile are the same. Applicable for ordinal, interval, and ratio data ot applicable for nominal data Stat 13D, UCLA, Ivo Dinov. 11 Percentiles: Computational Procedure Organize the data into an ascending ordered array. Calculate the P percentile location: i () n 1 Determine the percentile s location and its value. If i is a whole number, the percentile is the average of the values at the i and (i+1) positions. If i is not a whole number, the percentile is at the (i+1) position in the ordered array. Stat 13D, UCLA, Ivo Dinov. 1 Page

3 Percentiles: Example Raw Data: 14, 1, 19, 3,, 13, 8, 17 Ordered Array:, 1, 13, 14, 17, 19, 3, 8 Location of 3th percentile: 3 i 1 () 8 4. The location index, i, is not a whole number; i ; the whole number portion is 3; the 3th percentile is at the 3rd location of the array; the 3th percentile is 13. Stat 13D, UCLA, Ivo Dinov. 13 Quartiles Measures of central tendency that divide a group of data into four subgroups Q 1 : % of the data set is below the first quartile Q : % of the data set is below the second quartile Q 3 : 7% of the data set is below the third quartile Q 1 is equal to the th percentile Q is located at th percentile and equals the median Q 3 is equal to the 7th percentile Quartile values are not necessarily members of the data set Stat 13D, UCLA, Ivo Dinov. 14 Quartiles Q 1 Q Q 3 % % % % Quartiles: Example Ordered array:, 19, 114, 1, 11, 1, 1, 19 Q 1 + i Q () Q : + i Q () Q 3 : i Q () Stat 13D, UCLA, Ivo Dinov. 1 Stat 13D, UCLA, Ivo Dinov. Variability Variability o Variability in Cash Flow Variability Variability in Cash Flow o Variability Time Stat 13D, UCLA, Ivo Dinov. 17 Stat 13D, UCLA, Ivo Dinov. 18 Page 3

4 Measures of Variability: Ungrouped Data Measures of variability describe the spread or the dispersion of a set of data. Common Measures of Variability Range Interquartile Range Absolute Deviation Variance Standard Deviation Z scores Coefficient of Variation Range The difference between the largest and the smallest values in a set of data Simple to compute 3 41 Ignores all data points except the two extremes Example: Range Largest - Smallest Stat 13D, UCLA, Ivo Dinov. 19 Stat 13D, UCLA, Ivo Dinov. Interquartile Range Range of values between the first and third quartiles Range of the middle half Less influenced by extremes Interquartile Range Q3 Q1 Stat 13D, UCLA, Ivo Dinov. 1 Deviation from the Data set:, 9,, 17, 18 : X 6 µ 13 Deviations from the mean: -8, -4, 3, 4, Stat 13D, UCLA, Ivo Dinov. µ Absolute Deviation Average of the absolute deviations from the mean X X µ X µ M. A. D. X µ Population Variance Average of the squared deviations from the arithmetic mean X X µ ( ) X µ ( X µ ) Stat 13D, UCLA, Ivo Dinov. 3 Stat 13D, UCLA, Ivo Dinov. 4 Page 4

5 Population Standard Deviation Square root of the variance X X µ ( X µ ) ( X µ ) Stat 13D, UCLA, Ivo Dinov. Sample Variance Average of the squared deviations from the arithmetic mean X X X ( X X ),398 1,8 1,39 1,311 7, ( X X ) 39,6,41 4,76 13,4 663,866 n 1 663, , Stat 13D, UCLA, Ivo Dinov. 6 S Sample Standard Deviation Square root of the sample variance ( X X ) X X X ( X X ),398 1,8 1,39 1,311 7, ,6,41 4,76 13,4 663,866 n 1 663, , Stat 13D, UCLA, Ivo Dinov. 7 S S S 1, Uses of Standard Deviation Indicator of financial risk Quality Control construction of quality control charts process capability studies Comparing populations household incomes in two cities employee absenteeism at two companies Stat 13D, UCLA, Ivo Dinov. 8 Standard Deviation as an Indicator of Financial Risk Financial Security Annualized Rate of Return µ A 1% 3% B 1% 7% Empirical Rule Data are normally distributed (or approximately normal) Distance from the µ ± 1 µ ± µ ± 3 Percentage of Values Falling Within Distance Stat 13D, UCLA, Ivo Dinov. 9 Stat 13D, UCLA, Ivo Dinov. 3 Page

6 Chebyshev s Theorem Applies to all distributions P( µ k < X < µ + k) 1 for k>1 1 k Chebyshev s Theorem Applies to all distributions umber of Standard Deviations K K 3 Distance from the Minimum Proportion of Values Falling Within Distance µ ± 1-1/.7 µ ± 3 1-1/3.89 µ ± 4 K 4 1-1/4.94 Stat 13D, UCLA, Ivo Dinov. 31 Stat 13D, UCLA, Ivo Dinov. 3 Coefficient of Variation Ratio of the standard deviation to the mean, expressed as a percentage Measurement of relative dispersion CV.. ( ) µ 1 Coefficient of Variation µ µ 1 CV.. ( 1) ( 1) 1 CV.. 1 µ 84 1 µ. ( ) ( ) Stat 13D, UCLA, Ivo Dinov. 33 Stat 13D, UCLA, Ivo Dinov. 34 Measures of Central Tendency and Variability: Grouped Data Measures of Central Tendency Median Mode Measures of Variability Variance Standard Deviation Absolute Deviation of Grouped Data Weighted average of class midpoints Class frequencies are the weights µ fm f fm f 1M 1 + f M + f 3 M fim f 1 + f + f fi i Stat 13D, UCLA, Ivo Dinov. 3 Stat 13D, UCLA, Ivo Dinov. 36 Page 6

7 Calculation of Grouped Class Interval Frequency Class Midpoint fm -under under under under under under fm 1 µ f 43. Median of Grouped Data cf p Median L + ( W) fmed Where: L the lower limit of the median class cf p cumulative frequency of class preceding the median class f med frequency of the median class W width of the median class total of frequencies Stat 13D, UCLA, Ivo Dinov. 37 Stat 13D, UCLA, Ivo Dinov. 38 Median of Grouped Data -- Example Mode of Grouped Data Cumulative Class Interval Frequency Frequency -under under under under under under 8 1 cf p Md L + ( W ) fmed ( 1) Midpoint of the modal class Modal class has the greatest frequency Class Interval Frequency -under under under 11 -under under under Mode 3 Stat 13D, UCLA, Ivo Dinov. 39 Stat 13D, UCLA, Ivo Dinov. 4 Variance and Standard Deviation of Grouped Data Population Variance and Standard Deviation of Grouped Data Population f ( M µ ) S S S Sample f ( M X) n 1 Class Interval -under 3 3-under 4 4-under -under 6 6-under 7 7-under 8 f M ( µ ) fm M µ ( M µ ) f ( ) f M M µ Stat 13D, UCLA, Ivo Dinov. 41 Stat 13D, UCLA, Ivo Dinov. 4 Page 7

8 Measures of Shape Skewness Skewness Absence of symmetry Extreme values in one side of a distribution Kurtosis Peakedness of a distribution Leptokurtic: high and thin Mesokurtic: normal shape Platykurtic: flat and spread out Box and Whisker Plots Graphic display of a distribution Reveals skewness egatively (Left) Symmetric (ot ) Positively (Right) Stat 13D, UCLA, Ivo Dinov. 43 Stat 13D, UCLA, Ivo Dinov. Median egatively Mode Skewness Median Mode Symmetric (ot ) Mode Median Positively Coefficient of Skewness Summary measure for skewness If S <, the distribution is negatively skewed (skewed to the left). If S, the distribution is symmetric (not skewed). If S >, the distribution is positively skewed (skewed to the right). Stat 13D, UCLA, Ivo Dinov. 4 Stat 13D, UCLA, Ivo Dinov. Skewness & Kurtosis What do we mean by symmetry and positive and negative skewness? Kurtosis? Properties?!? 3 4 ( Yk Y ) ( Yk Y ) k 1 k 1 Skewness ; Kurtosis 3 4 ( 1) SD ( 1) SD Skewness in linearly invariant Sk(aX+b)Sk(X) Skewness is a measure of unsymmetry Kurtosis is a measure of flatness Both are use to quantify departures from Stdormal Skewness(Stdorm); Kurtosis(Stdorm)3 Stat 13D, UCLA, Ivo Dinov. 47 Kurtosis Peakedness of a distribution Leptokurtic: high and thin Mesokurtic: normal in shape Platykurtic: flat and spread out Leptokurtic Mesokurtic Platykurtic Stat 13D, UCLA, Ivo Dinov. 48 Page 8

9 Box and Whisker Plot Five secific values are used: Median, Q First quartile, Q 1 Third quartile, Q 3 Minimum value in the data set Maximum value in the data set Inner Fences IQR Q 3 -Q 1 Lower inner fence Q 1-1. IQR Upper inner fence Q IQR Outer Fences Lower outer fence Q 1-3. IQR Upper outer fence Q IQR Minimum Box and Whisker Plot Q 1 Q Q 3 Maximum Stat 13D, UCLA, Ivo Dinov. 49 Stat 13D, UCLA, Ivo Dinov. Skewness: Box and Whisker Plots, and Coefficient of Skewness S < S S > Pearson Product-Moment Correlation Coefficient egatively Symmetric (ot ) Positively r SSXY ( SSX )( SSY) ( X X)( Y Y) ( X X) ( Y Y) ( X)( Y) XY n ( X ) ( Y) X n Y n 1 r 1 Stat 13D, UCLA, Ivo Dinov. 1 Stat 13D, UCLA, Ivo Dinov. Three Degrees of Correlation r < r > r Computation of r for the Economics Example (Part 1) Futures Day Interest X Index Y X Y XY ,841 1, ,84 1, ,76 1, ,6 1, ,176 1, ,79 1, ,79 1, ,76 1, ,76 1, , 1, ,89 1, ,81 1,98. Summations 9.93, ,7 1,11.7 Stat 13D, UCLA, Ivo Dinov. 3 Stat 13D, UCLA, Ivo Dinov. 4 Page 9

10 Computation of r for the Economics Example (Part ) Scatter Plot and Correlation Matrix for the Economics Example ( X)( Y) XY r n ( X ) ( Y) X Y n n ( 9. 93)( 7) ( 111,. 7) 1 ( ) ( ) ( ) ( ) , Futures Index Interest Interest Futures Index Interest Futures Index Stat 13D, UCLA, Ivo Dinov. Stat 13D, UCLA, Ivo Dinov. 6 // Palindrome Testing Program (1 of ) // test for palindrome property Cstrings vs Strings! #include <iostream> #include <string> #include <cctype> using namespace std; void swap(char& lhs, char& rhs); //swaps char args corresponding to parameters lhs and rhs string reverse(const string& str); //returns a copy of arg corresponding to parameter //str with characters in reverse order. string removepunct(const string& src, const string& punct); //returns copy of string src with characters //in string punct removed string makelower (const string& s); //returns a copy of parameter s that has all upper case //characters forced to lower case, other characters unchanged. //Uses <string>, which provides tolower bool ispal(const string& this_string); //uses makelower, removepunct. //if this_string is a palindrome, // return true; //else // return false; 7 // Palindrome Testing Program ( of ) int main() string str; cout << "Enter a candidate for palindrome test " << "\nfollowed by pressing return.\n"; getline(cin, str); if (ispal(str)) cout << "\"" << str + "\" is a palindrome "; else cout << "\"" << str + "\" is not a palindrome "; cout << endl; return ; void swap(char& lhs, char& rhs) char tmp lhs; lhs rhs; rhs tmp; 8 // Palindrome Testing Program (3 of ) string reverse(const string& str) int start ; int end str.length(); string tmp(str); while (start < end) end--; swap(tmp[start], tmp[end]); start++; return tmp; //Returns arg that has all upper case characters forced to lower case, //other characters unchanged. makelower uses <string>, which provides //tolower string makelower(const string& s) //uses <cctype> string temp(s); //This creates a working copy of s for (int i ; i < s.length(); i++) temp[i] tolower(s[i]); return temp; 9 // Palindrome Testing Program (4 of ) //returns a copy of src with characters in punct removed string removepunct(const string& src, const string& punct) string no_punct; int src_len src.length(); int punct_len punct.length(); for(int i ; i < src_len; i++) string achar src.substr(i,1); int location punct.find(achar, ); //find location of successive characters //of src in punct if (location < location > punct_len) no_punct no_punct + achar; //achar not in punct -- keep it return no_punct; 6 Page 1

11 // Palindrome Testing Program ( of ) //uses functions makelower, removepunct. //Returned value: //if this_string is a palindrome, // return true; //else // return false; bool ispal(const string& this_string) string punctuation(",;:.?!'\" "); //includes a blank string str(this_string); str makelower(str); string lowerstr removepunct(str, punctuation); return lowerstr reverse(lowerstr); 61 Page 11