STATISTIC : DESCRIPTIVE MEASURES

1. STATISTIC : DESCRIPTIVE MEASURES Measures For Central Tendency Mean, Mode, Median ungrouped data Mean _ x = ∑x n Where n is the sample Mean µ = ∑x N Where N is the population

2. Mean _ x = ∑250 10 25 = STATISTIC : DESCRIPTIVE MEASURES Example : Mean The number of 911 calls classified as domestic disturbance calls in large metropolitan location where sampled for 10 randomly selected 24 hour periods with the following results. Find the mean number of calls per 24 hours period 10 20 50 30 20 40 20 30 10 20

3. ~ ~ x x ~ µ 20 + 20 = = 20 2 STATISTIC : DESCRIPTIVE MEASURES Median Median of a set of data is a value that divides the bottom 50% of the data from the top 50% of the data. To find the median of a data set, first arrange the data in increasing order. If the number of observations is odd then the median is the Number in the middle of the observation list. If the number is even then the median Is the mean of the two values closest to the middle of the ordered list = Population median = Sample median Example : Median The number of 911 calls classified as domestic disturbance calls in large metropolitan location where sampled for 10 randomly selected 24 hour periods with the following results. Find the median number of calls per 24 hours period 10 10 20 20 20 20 30 30 40 50 (even number of observation)

4. STATISTIC : DESCRIPTIVE MEASURES Mode The mode is the value in a data set that occurs the most often. If no such value exists, we say that the data has no mode. If two such values exist, we say the data is bimodal. If three such values exist we say the data set is trimodal. There is no symbol that is used to represent the mode. Data set : 10, 12, 15, 15, 18, 20 Mode: 15 Shapes of Distribution (i) Bell-shaped (ii) Left-skewed (iii) Right-skewed Bell-shaped

5. STATISTIC : DESCRIPTIVE MEASURES Shapes of Distribution Left-skewed Right-skewed

6. Mean = x = ∑xf n STATISTIC : DESCRIPTIVE MEASURES Measures For Central Tendency Mean, Mode, Median grouped data = where x represent the class marks, f represent the frequencies and n represent the sample size Mean = x = 9.5 x 7 + 19.5 x 15 + 29.5 x 5 + 39.5 x 5 + 49.5 x 5 37

7. Mode Spread evenly the number of size, size is 37 so the middle value should be 19 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37 Median size According to the table the median size would be at the class range 15-24. To be able to reach the median size we need to add 12 to the upper frequency Values from the above class range ( to reach 19) so median is calculated as Median = The lower boundary of class range + 12/15 x class size= 14.5 + (12/15) x 10)

8. Mode The modal class is defined to be the class with maximum frequency. The Mode for grouped data would be the class mark of the modal class. Mode = 19.5

9. STATISTIC : DESCRIPTIVE MEASURES Range, Variance, and Standard Deviation Ungrouped Data Range The range for a data set is equal to the maximum value in the data set minus the minimum value in the data set. Example : Range in test score for Johan and Jamal. Range for Johan is 100 – 85 = 15 and the range for Jamal is 90 – 60 = 30. The spread in Johan’s score is as measure by range twice the spread of Jamal’s score.

10. _ _ ∑( x – x )2 n-1 ∑( x – µ )2 N STATISTIC : DESCRIPTIVE MEASURES Range, Variance, and Standard Deviation Ungrouped Data Variance s2 = Variance for sample of size n δ2 = Variance for population of size N

11. STATISTIC : DESCRIPTIVE MEASURES Range, Variance, and Standard Deviation Ungrouped Data Variance: continue… Example : Times in minutes for 5 students to complete a task were 5, 10, 15, 3 and 7. The mean time is 8 minutes (refer back to mean). See Table 3.0, it illustrates the computation indicated by the formula variance for sample. _ _ _ _

12. 88 4 _ ∑( x – x )2 n-1 STATISTIC : DESCRIPTIVE MEASURES Range, Variance, and Standard Deviation Ungrouped Data Variance: continue… If we followed the variance formula in the previous slide, the variance for Table 3.0 is 22 minutes squared. s2 = = = 22 The standard deviation is then calculated as s = √s2 Sample standard deviation δ =√δ2 Population standard deviation The standard deviation is √22 = 4.7 minutes

13. µ δ STATISTIC : DESCRIPTIVE MEASURES Coefficient Of Variation The coefficient variation is equal to the standard deviation divided by the mean. The result is usually multiplied by 100 to express it as a percent. The coefficient of variation for a sample is given by CV = x 100% The coefficient of variation for population CV = x 100% s x

14. 6,125 20,100 X 100 = 30.5% 2,730 5,485 X 100 = 49.8% STATISTIC : DESCRIPTIVE MEASURES Coefficient Of Variation : continue… Example : A national sampling of prices for new and used cars found that the mean price for a new car is $20,000 and the standard deviation is $6,125 and that the mean price for a used car is $5,485 with a standard deviation equal to $2,730. In terms of absolute variation, the standard deviation of price for new cars is more than twice that of used cars. However, in terms of relative variation, there is more relative variation in the price of used cars than in new cars. The CV for used cars is and the CV for new cars is

15. Exercise : Answer all Question Q1 . Table below gives the selling prices in tens of thousands of dollars for 20 Homes sold during the past month. Find the mean, mode, and median. Mean = Mode= Median=

16. Q2 . Find the mean, mode, and median for the grouped data below Mean = Mode= Median=

17. _ ∑( x – x )2 n-1 Q3. Fill in the table below with the details required _ _ _ _ Variance s2 = =