Topic-1 Describing Data with Numerical Measures

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1 Topic-1 Describing Data with Numerical Measures

2 Central Tendency (Center) and Dispersion (Variability) Central tendency: measures of the degree to which scores are clustered around the mean of a distribution Dispersion: measures the fluctuations (variability) around the characteristics of central tendency

3 Measures of Center A measure along the horizontal axis of the data distribution that locates the center of the distribution.

4 Arithmetic Mean or Average The mean of a set of measurements is the sum of the measurements divided by the total number of measurements. x n å i 1 n x i where n number of measurements å x i sum of all the measurements

5 Example The set:, 9, 1, 5, 6 å x x i 6. 6 n 5 5 If we were able to enumerate the whole population, the population mean would be called m (the Greek letter mu ).

6 Median The median of a set of measurements is the middle measurement when the measurements are ranked from smallest to largest. The position of the median is 0.5(n + 1) once the measurements have been ordered.

7 Example The set :, 4, 9, 8, 6, 5, 3 n 7 Sort :, 3, 4, 5, 6, 8, 9 Position:.5(n + 1).5(7 + 1) 4 th Median 4 th largest measurement The set:, 4, 9, 8, 6, 5 n 6 Sort:, 4, 5, 6, 8, 9 Position:.5(n + 1).5(6 + 1) 3.5 th Median (5 + 6)/ 5.5 average of the 3 rd and 4 th measurements

8 Mode The mode is the measurement which occurs most frequently. The set:, 4, 9, 8, 8, 5, 3 The mode is 8, which occurs twice The set:,, 9, 8, 8, 5, 3 There are two modes 8 and (bimodal) The set:, 4, 9, 8, 5, 3 There is no mode (each value is unique).

9 Example The number of quarts of milk purchased by 5 households: Mean? x å x n Median? i m Mode? (Highest peak) mode Relative frequency 10/5 8/5 6/5 4/5 / Quarts 3 4 5

10 Extreme Values The mean is more easily affected by extremely large or small values than the median. The median is often used as a measure of center when the distribution is skewed.

11 Extreme Values Symmetric: Mean Median Skewed right: Mean > Median Skewed left: Mean < Median

12 Measures of Variability A measure along the horizontal axis of the data distribution that describes the spread of the distribution from the center. Range üdifference between maximum and minimum values Interquartile Range üdifference between third and first quartile (Q 3 -Q 1 ) Variance üaverage * of the squared deviations from the mean Standard Deviation üsquare root of the variance Definitions of population variance and sample variance differ slightly.

13 The Range The range, R, of a set of n measurements is the difference between the largest and smallest measurements. Example: A botanist records the number of petals on 5 flowers: 5, 1, 6, 8, 14 The range is R Quick and easy, but only uses of the 5 measurements.

14 Percentile 50 th Percentile 5 th Percentile 75 th Percentile º Median (Q ) º Lower Quartile (Q 1 ) º Upper Quartile (Q 3 ) Interquartile Range: IQRQ 3 Q 1

15 The position of p-th percentile is 0.p(n + 1) The position of Q 1 is The position of Q 3 is 0.5(n + 1) 0.75(n + 1) once the measurements have been ordered. If the positions are not integers, find the quartiles by interpolation.

16 Example The prices ($) of 18 brands of walking shoes: Position of Q 1 0.5(18 + 1) 4.75 Position of Q (18 + 1) 14.5 üq 1 is 3/4 of the way between the 4 th and 5 th ordered measurements, or Q (65-65) 65.

17 Example The prices ($) of 18 brands of walking shoes: Position of Q 1 0.5(18 + 1) 4.75 Position of Q (18 + 1) 14.5 üq 3 is 1/4 of the way between the 14 th and 15 th ordered measurements, or Q (75-74) 74.5 üand IQR Q 3 Q

18 90-th percentile P 90 The position of 90-th percentile is 0.9(18 + 1)17.1 The prices ($) of 18 brands of walking shoes: P (95-90) 90.5

19 The Variance The variance is measure of variability that uses all the measurements. It measures the average deviation of the measurements about their mean. Flower petals: 5, 1, 6, 8, x 9 5 x

20 The Variance The variance of a population of N measurements is the average of the squared deviations of the measurements about their mean m. s å ( - m N x i ) The variance of a sample of n measurements is the sum of the squared deviations of the measurements about their mean, divided by (n 1). s å ( x i - x ) n - 1

21 The Standard Deviation In calculating the variance, we squared all of the deviations, and in doing so changed the scale of the measurements. To return this measure of variability to the original units of measure, we calculate the standard deviation, the positive square root of the variance. Population standard deviation : s s Sample standard deviation : s s

22 Two Ways to Calculate the Sample Variance Use the Definition Formula: xi x i Sum x ( x - x) i s s s å( xi n x)

23 Two Ways to Calculate the Sample Variance Use the calculation formula: x i x i Sum s s ( å xi ) å xi - n n s

24 Using Measures of Center and Spread: The Empirical Rule Given a distribution of measurements that is approximately mound-shaped: üthe interval m ±scontains approximately 68% of the measurements. üthe interval m ± s contains approximately 95% of the measurements. üthe interval m ± 3s contains approximately 99.7% of the measurements.

25 The Empirical Rule: An Example

26 Approximating s From Chebysheff s Theorem and the Empirical Rule, we know that R» 4-6 s To approximate the standard deviation of a set of measurements, we can use: s» R / 4 or s» R / 6 for a large dataset.

27 Approximating s The ages of 50 tenured faculty at a state university R s» R / 4 44/ 4 11 Actual s 10.73

28 Measures of Relative Standing Where does one particular measurement stand in relation to the other measurements in the data set? How many standard deviations away from the mean does the measurement lie? This is measured by the z-score. z - score x - s x Suppose s. s 4 s s x 5 x 9 x 9 lies z std dev from the mean.

29 z-scores z-scores between and are not unusual. z-scores should not be more than 3 in absolute value. z-scores larger than 3 in absolute value would indicate a possible outlier. Outlier Not unusual Outlier z Somewhat unusual

30 Constructing a Box Plot (Refer Lecture -1) ücalculate Q 1, the median (Q ), Q 3 and IQR. üdraw a horizontal line to represent the scale of measurement. üdraw a box using Q 1, the median, Q 3. Q 1 m Q 3

31 Constructing a Box Plot cont. üisolate outliers by calculating ülower fence: Q IQR üupper fence: Q IQR ümeasurements beyond the upper or lower fence is are outliers and are marked (*). * LF Q 1 m Q 3 UF

32 Example Amt of sodium in 8 brands of cheese: Q m 35 Q m Q 1 Q 3

33 Example IQR Lower fence (47.5) 1.5 Upper fence (47.5) Outlier: x 50 * m LF Q 1 Q 3 UF

34 Grouped and Ungrouped Data

35 Sample Mean Ungrouped data: x n x x + x + i å 1 1 K n n i + x n n number of observation Grouped data: x k å * fi xi k i 1 1 æ çå n n è i 1 f i x * i ö ø n sum of the frequencies k number class f i frequency of each class * x i midpoint of each class

36 Example: - ungrouped data Resistance of 5 coils: 3.35, 3.37, 3.8, 3.34, 3.30 ohm. The average: x n åxi i n 5

37 Example: - grouped data Frequency Distributions of the life of 30 tires in 1000 km * Boundaries Midpoint Frequency, f i i x * k Total n 30 i i å i 1 f i x i x k å fix i 1 f n 11, * i

38 Sample Variance Ungrouped data: 1 1 ) ( ø ö ç è æ å å å n n x x n x x s n i i n i i n i i Grouped data: 1) ( ) ( 1 1 * * - ø ö ç è æ - å å n n x f x f s k i k i i i i i

39 Example- ungrouped data Sample: Moisture content (%) of kraft paper are: 6.7, 6.0, 6.4, 6.4, 5.9, and 5.8. s (31.6) - (37.) (6-1) Sample standard deviation, s 0.35 %

40 Calculating the Sample Standard Deviation - Grouped Data Standard deviation for a grouped sample: Table: Car speeds in km/h Boundaries * xi fi * f i x i * f i x i s k * ( f ixi i 1 å k æ ) - çå fix è i 1 ( n -1) (90,74) - (9354) 96 Total s 9. 9 (96-1) * i ö ø n

41 Mode Grouped Data For grouped data, class mode (or, modal class) is the class with the highest frequency. Where: d d l Mode L + d 1 d + 1 L Lower boundary of the modal class frequency of modal class frequency of the pre modal class 1 d frequency of modal class frequency of the post modal class length of the modal class l

42 Example of Grouped Data (Mode) Based on the grouped data below, find the mode Class Limit Boundaries f Cum Freq Cum Rel Freq Modal Class (third) L l 10 d d Mode 1 L + l d d + d

43 Percentile (Ungrouped Data) The p-th percentile class from the frequency table: k (n x p)/100 where n total frequency and p is the p-th percentile Find the p-th class in which the value of k lies in the corresponding cumulative frequency group.

44 Percentile (Grouped Data) Sp LBCp + k - CFB f p l k n p 100 n number of observations LBCp Lower Boundary of the percentile class CFB Cumulative frequency of the class before Cp l Class width fp Frequency of percentile class

45 Example- Percentile for grouped data Using the previous table of freq, find the median and 80 th percentile? Class Limit Boundaries f Cum Freq Cum Rel Freq

46 Median, S50 k C 50 LBC p (119 18) (3 rd class) CFB 13 l f p S 50 æ 0-13 ö ç 10 è 13 ø th percentile, S80 k C 80 LBC p S (19 138) (4 rd class) 18.5 CFB 6 l 10 8 f p æ 3-6 ö ç 10 è 8 ø 136

47 Coefficient of Variation(CV) When comparing between data sets with different units or widely different means, one should use the coefficient of variation for comparison instead of the standard deviation. The Coefficient of Variation can be written as CV s x We express CV as a percentage by multiplying 100 Less than 30% consider good