Using Measures of Center And Spread Tchebysheff’s Theorem

USING MEASURES OF CENTER AND SPREAD: TCHEBYSHEFF’S THEOREM Given a number k greater than or equal to 1 and a set of n measurements, at least 1-(1/k2) of the measurement will lie within k standard deviations of the mean.  Can be used for either samples ( x and s) or for a population (m and s). Important results: If k = 2, at least 1 – 1/22 = 3/4 of the measurements are within 2 standard deviations of the mean. If k = 3, at least 1 – 1/32 = 8/9 of the measurements are within 3 standard deviations of the mean. USING MEASURES OF CENTER AND SPREAD: THE EMPIRICAL RULE Given a distribution of measurements that is approximately mound-shaped: The interval m  s contains approximately 68% of the measurements. The interval m  2s contains approximately 95% of the measurements. The interval m  3s contains approximately 99.7% of the measurements. EXAMPLE The ages of 50 tenured faculty at a state university.    34 42 34 43 48 31 59 50 70 36 34 30 63 48 66 43 52 43 40 32 52 26 59 44 35 58 36 58 x  44.9 s  10.73 50 37 43 53 43 52 44 62 49 34 48 53 39 45 41 35 36 62 34 38 28 53 14/50 12/50 Relative frequency  10/50 8/50 6/50 4/50 2/50 Shape? Skewed right 0 25 33 41 49 Ages 57 65 73 k x ks Interval Proportion in Interval Tchebysheff Empirical Rule 1 44.9 10.73 34.17 to 55.63 31/50 (.62) At least 0  .68 2 44.9 21.46 23.44 to 66.36 49/50 (.98) At least .75  .95 3 44.9 32.19 12.71 to 77.09 50/50 (1.00) At least .89  .997 •Do the actual proportions in the three intervals agree with those given by Tchebysheff’s Theorem? •Yes. Tchebysheff’s Theorem must be true for any data set. •Do they agree with the Empirical •No. Not very well. Rule? • The data distribution is not •Why or why not? very mound-shaped, but skewed right. EXAMPLE The length of time for a worker to complete a specified operation averages 12.8 minutes with a standard deviation of 1.7 minutes. If the distribution of times is approximately mound-shaped, what proportion of workers will take longer than 16.2 minutes to complete the task? 95% between 9.4 and 16.2 47.5% between 12.8 and 16.2 .47 .47 5 5 .025 (50-47.5)% = 2.5% above 16.2 APPROXIMATING S  From Tchebysheff’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 largedata set. APPROXIMATING S The ages of 50 tenured faculty at a state university. • • • • 34 42 34 43 48 31 59 50 70 36 34 30 63 48 66 43 52 43 40 32 52 26 59 44 35 58 36 58 50 37 43 53 43 52 44 62 49 34 48 53 39 45 41 35 36 62 34 38 28 53 R = 70 – 26 = 44 s  R / 4  44 / 4  11 Actual s = 10.73 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. xx z - score  s Suppose s = 2. s 4 s x 5 s x9 x = 9 lies z =2 std dev from the mean. Z-SCORES From Tchebysheff’s Theorem and the Empirical Rule  At least 3/4 and more likely 95% of measurements lie within 2 standard deviations of the mean.  At least 8/9 and more likely 99.7% of measurements lie within 3 standard deviations of the mean.  z-scores between –2 and 2 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 Outlier Not unusual z -3 3 -2 -1 0 Somewhat unusual 1 2 MEASURES OF RELATIVE STANDING  How many measurements lie below the measurement of interest? This is measured by the pth percentile. p% (100-p) % p-th percentile x EXAMPLES  90% of all men (16 and older) earn more than $319 per week. BUREAU OF LABOR STATISTICS 10% 90% $319 $319 is the 10th percentile. 50th Percentile  Median 25th Percentile  Lower Quartile (Q1) 75th Percentile  Upper Quartile (Q3) QUARTILES AND THE IQR • The lower quartile (Q1) is the value of x which is larger than 25% and less than 75% of the ordered measurements. • The upper quartile (Q3) is the value of x which is larger than 75% and less than 25% of the ordered measurements. • The range of the “middle 50%” of the measurements is the interquartile range, IQR = Q3 – Q1 CALCULATING SAMPLE QUARTILES • The lower and upper quartiles (Q1 and Q3), can be calculated as follows: • The position of Q1 is .25(n + 1) •The position of Q3 is .75(n + 1) once the measurements have been ordered. If the positions are not integers, find the quartiles by interpolation. EXAMPLE The weights of 18 patients in kgs: 40 60 65 65 65 68 68 70 70 70 70 70 70 74 75 75 90 95 Position of Q1 = .25(18 + 1) = 4.75 Position of Q3 = .75(18 + 1) = 14.25 Q1is 3/4 of the way between the 4th and 5th ordered measurements, or Q1 = 65 + .75(65 - 65) = 65. EXAMPLE The weights of 18 patients in kgs: 40 60 65 65 65 68 68 70 70 70 70 70 70 74 75 75 90 95 Position of Q1 = .25(18 + 1) = 4.75 Position of Q3 = .75(18 + 1) = 14.25 Q3 is 1/4 of the way between the 14th and 15th ordered measurements, or Q3 = 74 + .25(75 - 74) = 74.25 and IQR = Q3 – Q1 = 74.25 - 65 = 9.25 USING MEASURES OF CENTER AND SPREAD: THE BOX PLOT The Five-Number Summary: Min Q1 Median Q3 Max •Divides the data into 4 sets containing an equal number of measurements. •A quick summary of the data distribution. •Use to form a box plot to describe the shape of the distribution and to detect outliers. CONSTRUCTING A BOX PLOT Calculate Q1, the median, Q3 and IQR. Draw a horizontal line to represent the scale of measurement. Draw a box using Q1, the median, Q3. Q1 m Q3 CONSTRUCTING A BOX PLOT Isolate outliers by calculating Lower fence: Q1-1.5 IQR Upper fence: Q3+1.5 IQR Measurements beyond the upper or lower fence is are outliers and are marked (*). * Q1 m Q3 CONSTRUCTING A BOX PLOT Draw “whiskers” connecting the largest and smallest measurements that are NOT outliers to the box. * Q1 m Q3 EXAMPLE Amount of sodium in 8 brands of cheese: 260 290 300 320 330 340 340 520 Q1 = 292.5 m = 325 m Q1 Q3 Q3 = 340 EXAMPLE IQR = 340-292.5 = 47.5 Lower fence = 292.5-1.5(47.5) = 221.25 Upper fence = 340 + 1.5(47.5) = 411.25 Outlier: x = 520 * m Q1 Q3 INTERPRETING BOX PLOTS  Median line in center of box and whiskers of equal length—symmetric distribution  Median line left of center and long right whisker—skewed right  Median line right of center and long left whisker—skewed left KEY CONCEPTS I. Measures of Center 1. Arithmetic mean (mean) or average a. Population: m b. Sample of size n:x   xi n 2. Median: position of the median  .5(n +1) 3. Mode 4. The median may preferred to the mean if the data are highly skewed. II. Measures of Variability 1. Range: R  largest  smallest KEY CONCEPTS 2. Variance ( xi  m ) 2 2 a. Population of N measurements: s  N b. Sample of n measurements: ( xi ) 2  xi  2  ( x  x ) n i s2   n 1 n 1 2 3. Standard deviation Population standard deviation : s  s 2 Sample standard deviation : s  s 2 4. A rough approximation for s can be calculated as s  R / 4. The divisor can be adjusted depending on the sample size. KEY CONCEPTS III. Tchebysheff’s Theorem and the Empirical Rule 1. Use Tchebysheff’s Theorem for any data set, regardless of its shape or size. a. At least 1-(1/k 2 ) of the measurements lie within k standard deviation of the mean. b. This is only a lower bound; there may be more measurements in the interval. 2. The Empirical Rule can be used only for relatively moundshaped data sets.  Approximately 68%, 95%, and 99.7% of the measurements are within one, two, and three standard deviations of the mean, respectively. KEY CONCEPTS IV. Measures of Relative Standing 1. Sample z-score: 2. pth percentile; p% of the measurements are smaller, and (100  p)% are larger. 3. Lower quartile, Q 1; position of Q 1  .25(n +1) 4. Upper quartile, Q 3 ; position of Q 3  .75(n +1) 5. Interquartile range: IQR  Q 3  Q 1 V. Box Plots 1. Box plots are used for detecting outliers and shapes of distributions. 2. Q 1 and Q 3 form the ends of the box. The median line is in the interior of the box. KEY CONCEPTS 3. Upper and lower fences are used to find outliers. a. Lower fence: Q 1  1.5(IQR) b. Upper fence: Q 3 + 1.5(IQR) 4. Whiskers are connected to the smallest and largest measurements that are not outliers. 5. Skewed distributions usually have a long whisker in the direction of the skewness, and the median line is drawn away from the direction of the skewness.