DESCRIPTIVE STATISTICS2 Lectures delivered by Prof K K

DESCRIPTIVE STATISTICS-2 Lectures delivered by Prof. K. K. Achary YENEPOYA RESEARCH CENTRE

MEDIAN • The median of a given data set is the value which divides the ordered set of values into two equal parts such that the number of values equal to or greater than the median is equal to the number of values less than or equal the median. • In a sample data set having n values , the median is (n+1)/2 th value, in the ordered sample. • CAUTION: Median is not (n+1)/2!!! • Note that when n is odd the median is the middlemost value.

• If n =11 then median is the sixth observation. • If n=12 then the median is the value at 6. 5 th position. • Ex: age data 22, 24, 23, 21, 25, 26, 23, 24, 25, 26, 23, 24 • Arrange in increasing order: • 21, 22, 23, 23, 24, 24, 24, 25, 26, 26

• • • n=15. Hence position of the median is (15+1)/2=8 median=8 th observation= 24 The median age at first childbirth is 24 years. ex: 34, 45, 67, 87, 39, 93, 56, 48 (ages of persons ) N=8 & median is 4. 5 th value Ordered sample : 34, 39, 45, 48, 56, 67, 87, 93 Median is average of 4 th and 5 th values Median=(48+56)/2=104/2=52 years What is the mean age in the above cases?

• Compute the mean age for the sample. • Mean age is 58. 63 years • Mean is large here due to the influence of large values • Hence median is better in such cases • Computation of median in grouped data: – For discrete frequency table use the definition and identify the value corresponding to the cum. freq (n+1)/2

• Ex • Score level (xi) • 0 • 1 • 2 • 3 • 4 • 5 (fi) 3 12 23 46 18 6 cum. freq. 3 15 38 84 102 108

n=108 & (n+1)/2=54. 5= position of median Hence median is the 54. 5 th value in the ordered sample. Observe that the 54 th and 55 th observations are having the value 3. Hence median score is 3.

Activity : Find the mean and median of the following data( fbs(mg/dl) ) • 94 86 180 252 143 168 77 270 134 86 99 285 200 99 118 117 101 149 137 84 139 145 108 126 108 69 264

Computation of median from grouped data(continuous freq. table ) • First compute the true class limits(if required ) and the cumulative frequency of each class • Identify the first class interval for which the cumulative frequeny is greater than or equal to (n/2). This class is the median class. • Use the following formula to compute the median.

Median • • • Here the notations have the following meaning L = lower limit of the median class i = width of the median class f= frequency of the median class n= total no. of observations ( sample size) c= cum. freq. of class preceding the median class

Example Class interval Freq. Cum. freq. 5 - 10 3 3 10 15 20 25 - 15 20 25 30 8 16 12 9 11 27 39 48 30 35 - 35 40 6 6 54 60

• Here n= 60, so n/2 = 30 • Hence median class is 20 – 25, L=20 & i=5 • f=12 • c=27 • So median = 20 + (5/12)(30 -27) • median = 21. 25 50% of the observations fall below 21. 25 and 50% of the observations are greater than or equal the median value.

• HW: Compute mean for the above data and compare with median. What are your observations? • Properties of median: • It is unique. It is computed from the ordered data. It is a positional average. It is not influenced by extreme values. If the extreme class intervals are open ended, then also we can compute median.

MODE • Mode is the most commonly occurring value in a data set. • For ungrouped data mode can be obtained by finding the value which appears /repeats max. no. of times. • Ex: Consider the data set: • 3, 4, 3, 5, 5, 6, 4, 3, 5, 5, 7, 8, 8, 5, 8, 6, 9, 5 • Here 5 repeats 6 times. Hence mode=5

• Note: Mode is not unique. In a dataset more than one value might repeat max. time. • Ex: Consider the data set: • 3, 4, 3, 5, 5, 6, 4, 3, 5, 5, 3, 8, 8, 5, 3, 6, 3, 5 • Here 3 and 5 repeat 6 times. Hence both 3 and 5 are modal values. • Question: If in a data set all values repeat equal no. of times, what is the mode?

Grouped discrete frequency distribution • In a grouped frequency table, mode is identified as the value with the largest frequency. • Score level (xi) (fi) • 0 3 • 1 12 • 2 23 • 3 46 • 4 18 • 5 6 • Mode=3

Computing mode in continuous frequency distribution • Mode is the value that has the highest frequency in a data set. • For grouped data, modal class is the class with the highest frequency. • To find mode for grouped data, use the following formula:

• Here L = lower limit/boundary of modal class • i = width of modal class • ∆1= difference between the frequency of • modal class and the class before the modal class • ∆2 = difference between the frequency of modal class and the frequency of the class after the modal class

• The above formula can be written in the following form: • Mode= L= lower limit/boundary of modal class i= width of modal class f 0 = freq. of class preceding the modal class f 1 = freq. of modal class f 2 = freq. of class next to modal class Note: compare with the previous formula ∆1 = f 1 -f 0, ∆2 = f 1 - f 2

• Summary & comments Descriptive statistics are summary figures or measures to summarise a data set using a single figure/ number. The measures of central tendency provide a summary statistic giving an idea about the concentration of the data around a central value. Mean, median and mode are three important measures of central tendency. Use mean for data in interval or ratio scale and median for ordinal scale. Mode is valid in all cases but it is an ill-defined measure. While mean is used in parametric inference, median is an important measure used in non-parametric inference.

• Median can be computed graphically from the cumulative frequency curves/ogive curve

• Some remarks: • While computing mean, median and mode for continuous data from grouped frequency tables, we are not using the original sample values. (this information is lost when we group the data). • To compute mean we assume that the values in each class are concentrated at the midpoint. Hence we are getting an estimate of the mean. • To compute median and mode, we use interpolation technique to estimate the measures. • Hence , in grouped data we estimate the measures of central tendency.

Partition values • Partition values are measures computed from ordered data. These are defined with reference to the position of the measure. The important measures are: • Quartiles: These are measures which divide the ordered data into four equal parts. The first quartile, is the value which divides the ordered dataset in such a way that 25% of the observations fall below it and the remaining observations lie above it. In an ungrouped data, is the value at the position • The second quartile is the median. • The third quartile is the value which divides the ordered dataset such that 75% of the observations fall below it and the remaining observations lie above it. It is denoted by. and located at the position

Example • Score level (xi) • 0 • 1 • 2 • 3 • 4 • 5 (fi) 3 12 23 46 18 6 cum. freq. 3 15 38 84 102 108

• • • Position of is (n+1)/4 = 109/4 =27. 25 Hence the first quartile value is 2. Position of third quartile = 3(n+1)/4 = 81. 75 Hence third quartile value is 3. _______________ Workout more examples

• For grouped continuous data

Age No. of patients(frequency ) Cum. freq. 10 - 19 4 4 20 - 29 66 70 30 - 39 47 117 40 - 49 36 153 50 - 59 12 165 60 – 69 4 169 Total 169

• • Here n/4 = 169/4 = 42. 25 First quartile class is 20 – 29 L = 19. 5 , i = 10 , f = 66, c = 4 Q 1 = 19. 5 + (10/66) ( 42. 25 – 4 ) =25. 30 3 n/4 = 126. 75. Q 3 class = 40 – 49 L=39. 5, i=10, f=36, c=117 Q 3 = 39. 5 +(10/36) ( 126. 75 – 117)=41. 94

Percentiles • Self study • Prepare a study report on percentiles– definition, computation, application and interpretation.