CHAPTER 2. Mean This is the usual arithmetic mean or average and is equal to the sum of the measurements divided by number of measurements.

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1 CHAPTER 2 umerical Measures Graphical method may ot always be sufficiet for describig data. You ca use the data to calculate a set of umbers that will covey a good metal picture of the frequecy distributio. umerical descriptive measures associated with a populatio of measuremet are called parameters; those computed from sample measuremets are called statistics. Measures of Locatio Mea This is the usual arithmetic mea or average ad is equal to the sum of the measuremets divided by umber of measuremets. Sample mea = X = i=1 X i Populatio mea = µ = i=1 X i Media This is the middle of the measuremets whe ordered them. The positio of the media = Mode The mode is measuremet which occurs most frequetly. ote: Mea ad media are equal whe distributio of data is symmetric, mea is greater whe distributio is skewed to right ad is less tha media whe distributio is skewed to left. Example: The prices for 14 differet brads of water-packed light tua are 0.99, 1.92,1.23, 0.85, 0.65, 0.53, 1.41, 1.12, 0.63, 0.67, 0.69, 0.60, 0.60, 0.66 a. Fid the average price for the 14 differet brads of tua. b. Fid the media price for the 14 differet brads of tua. 1

2 c. Based o your fidigs i parts a ad b, do you thik that the distributio of prices is skewed? - Measures of Variability Data sets may have the same ceter but look differet because of the way the umbers spread out from the ceter. Measures of variability ca help you create a metal picture of the spread of data. Rage R=largest measuremet - smallest measuremet Variace It measures the average deviatio of the measuremets about their mea Populatio variace : σ 2 = i=1 (X i µ) 2 = i=1 X2 i ( i=1 X i) 2 Sample variace : s 2 = i=1 (X i X) 2 1 = i=1 X2 i ( i=1 X i) 2 1 ote: Xi 2 = sum of squares of measuremets ad ( X i ) 2 = square of the sum of measuremets., Stadard deviatio Populatio stadard deviatio : σ = σ 2 Sample stadard deviatio : s = s 2 Example: You are give = 8 measuremets: 3, 1, 5, 6, 4, 4, 3, 5. a. Calculate the rage. b. Calculate the sample mea. c. Calculate the sample variace ad stadard deviatio. d. Compare the rage ad the stadard deviatio. The rage is approximately how may stadard deviatios? 2

3 Approximately R 4s or s R 4 Usig Measures of Ceter ad Spread, Tchebysheff s Theorem Give a umber k greater tha or equal to 1 ad a set of measuremets, at least 1 ( 1 k 2 ) of the measuremet will lie withi k stadard deviatios of the mea. Ca be used for either samples ( X ad s) or for a populatio (µ ad σ). Importat Result: If k = 2, at least 1 1/2 2 = 3/4 of the measuremets are withi 2 stadard deviatios of the mea. If k = 3, at least 1 1/3 2 = 8/9 of the measuremets are withi 3 stadard deviatios of the mea. Usig Measures of Ceter ad Spread, The Empirical Rule Give a distributio of measuremets that approximately moud-shaped: The iterval µ ± σ cotais approximately 68% of the measuremets. The iterval µ ± 2σ cotais approximately 95% of the measuremets. The iterval µ ± 3σ cotais approximately 99.7% of the measuremets. Examples: The ages of 50 teured faculty at a state uiversity are 34, 48, 70, 63, 52, 52, 35, 50, 37, 43, 53, 43, 52, 44, 42, 31, 36, 48, 43, 26, 58, 62, 49, 34, 48, 53, 39, 45, 34, 59, 34, 66, 40, 59, 36, 41, 35, 36, 62, 34, 38, 28, 43, 50, 30, 43, 32, 44, 58, Do the data agree with those give by Tchebysheff s Theorem? 2. Do they agree with the Empirical Rule? Why? The legth of time for a worker to complete a specified operatio averages 12.8 miutes with a stadard deviatio of 1.7 miutes. If the distributio of times is approximately moud-shaped, what proportio of workers will take loger tha 16.2 miutes to complete the task? 3

4 Measures of Relative Stadig Where does oe particular measuremet stad i relatio to the other measuremets i the set of data? z-score How may stadard deviatios away from the mea does the measuremet lie? This is measured by the z-score. The sample z-score defied by z score = x x s z-score betwee -2 ad 2 are ot uusual. z-score should ot be more tha 3 i absolute value. z-scores larger tha 3 i absolute value would idicate a possible outlier. Percetiles How may measuremets lie below the measuremet of iterest? This is measured by p th percetile. p th percetile is the value of measuremet that is more tha p% of the measuremets i ordered data. Quartiles Lower quartile (first quartile): 25 th percetile. (Q 1 ), is the value of x which is larger tha 25% ad less tha 75% of the ordered measuremets. The positio of the first quartile = 0.25( + 1) Upper quartile (third quartile): 75 th percetile. (Q 3 ), is the value of x which is larger tha 75% ad less tha 25% of the ordered measuremets. The positio of the third quartile = 0.75( + 1) It is obvious that secod quartile is media which i the other had is 50 th percetile. ote that, if the positio of quartile are ot a iteger, eed some modificatio. Iterquartile rage: The rage of the middle 50 th of the measuremets IQR = Q 3 Q 1 Examples: The prices of 18 brads of walkig shoes: 50, 60, 65, 65, 65, 68, 68, 70, 70, 70, 70, 70, 70, 74, 75, 75, 90, 95 Fid IQR ad media. 4

5 Box plot Box plot describes ceter of data, how spread the data, the exted ad ature of ay departure from symmetry, ad idetificatio of outliers. I geeral, box plot is based o five umber summary: Smallest value, first quartile, media, third quartile, largest value. Costructig box plot Calculate five umber summary ad also the IQR. Show five umbers o horizotal lie ad draw a box above the horizotal lie from Q 1 to Q 3 ad determie media by a vertical lie through the box. Draw 2 vertical lies from lower fece ad upper fece Lower fece = Q 1 1.5(IQR) Upper fece = Q (IQR) Determie the outliers (ay observatio beyod the feces) by. Draw two horizotal lies from the ed of the box to largest ad smallest observatios which are ot outliers (whiskers). Iterpretig Box plot Media lie i ceter of box ad whiskers od equal legth- symmetric distributio Media lie left of ceter ad log right whisker- skewed right Media lie right of ceter ad log left whisker- skewed left - Example: Costruct a box plot for these data ad idetify ay outliers: 25, 22, 26, 23, 27, 26, 28, 18, 25, 24, 12 - Suggested Exercises: 2.5, 2.11, 2.13, 2.15, 2.17, 2.19, 2.21, 2.27, 2.31, 2.35, 2.43, 2.49, 2.53, 2.57, 2.63, 2.65,

Chapter 2 Descriptive Statistics

Chapter 2 Descriptive Statistics Statistics Most commoly, statistics refers to umerical data. Statistics may also refer to the process of collectig, orgaizig, presetig, aalyzig ad iterpretig umerical data