Analyzing Data Finding centers of data set, describing variation of data set, and a shape of data set.

Transcription

1 Chapter : Describing Distributions with umbers Descriptive Statistics Describe the important characteristics of a set of measurements. Analyzing Data Finding centers of data set, describing variation of data set, and a shape of data set. Two Basic Concepts of Measures of Center Mean (x) (Arithmetic Mean) / (An average) : Found by adding the data values and dividing the total by the number of data. Sample mean x = n Population mean µ = Median(M): The middle of value when the original data values are arranged in order of increasing (or decreasing). (A center of an ordered data) Round-off Rule: Carry one more decimal place than is present in the original set of values. Ex. 1 17, 19, 1, 18, 0, 18, 19, 0, 0 Ex. 17, 19, 1, 18, 0, 18, 19, 0, 0, 1 Comparing the mean and median The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median.

2 Percentiles: The position measures used in educational and health-related fields to indicated the position of an individual in a group. P % (100 P )% -+ P th percentiles Median: = P 50 = Q The 50th percentile, denoted P 50, has about 50% of the data values below it and about 50% of the data value above it. Measuring variability: the quartiles First quartile (Q 1 ) = also called the lower quartile or the 5th percentile. P 5 Second quartile (Q ) = also called the median or the 50th percentile. P 50 Third quartile (Q 3 ) = also called the upper quartile or the 75th percentile. P 75 Measuring variability: The five-number summary and boxplots Boxplot: a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1, the median, and the third quartile, Q3. 5-umber Summary and Boxplot 1. Minimum data value. First quartile (Q 1 )= P 5 : At least 5% of the sorted values are less than or equal to Q 1, and at least 75% of the values are greater than or equal to Q Second quartile (Q )= P 50 : Same as the median; separates the bottom 50% of the sorted values from the top 50%. 4. Third quartile (Q 3 )= P 75 : At least 75% of the sorted values are less than or equal to Q 3, and at least 5% of the values are greater than or equal to Q Maximum data value

3 When the rth number that is a Q 1, Q, Q 3, r satisfies r total number of values r total number of values r total number of values = 0.5 r = 0.5 (total number of values) = 0.50 r = 0.50 (total number of values) = 0.75 r = 0.75 (total number of values) If r is a whole number: The value of the rth percentile is the midway between the rth and the (r + 1)th value. If r is not a whole number: Round up to the next larger whole number. Use the rth value. Ex. Construct Boxplot of the data set: 34, 36, 39, 43, 51, 53, 6, 63, 73, 79 Minimum data value 34 Q 1 = P 5 r 10 = 0.5 r =.5 the 3 rd value 39 Q = P 50 r 10 = 0.50 r = 5 the value between 5 th and 6 th = 5 Q 3 = P 75 r 10 = 0.75 r = 7.5 the 8 th value 63 Maximum data value 79

4 Spotting suspected outliers (Median as the center) Using a median and the Interquartile Range (IQR) to analyze data. Interquartile Range (IQR) : (Q 3 Q 1 ) Outliers with IQR Lower fence: Q (IQR) Upper fence: Q (IQR) Measuring spread: Variance and the Standard Deviation Those tools show the characteristic of data s variation. Range = (maximum data value) (minimum data value) Variance (s ): The average of the squares of the distance each value is from the mean. Standard Deviation (s): A measure of how much data values deviate away from the mean. The square root of the variance. A.M ean V ariance Standard Deviation Sample x = n s = (x x) n 1 s = (x x) n 1 Population µ = σ = (x µ) σ = (x µ)

5 Ex. 5, 7, 1,, 4 Range = (maximum data value) (minimum data value) = 7 1 = 6 Mean (x) = n = ( ) 5 = 19 5 = 3.8 Steps Step 1: Compute the mean x. Step : Subtract the mean from each individual value (x - x). Step 3: Square each of the deviations obtain from Step. (x x). Step 4: Add all of the squares obtained from Step 3. (x x) Step 5: Divided the total from Step 4 by the number n 1, which is 1 less than the total number of sample values present. The result is the variance. Step 6: Find the square root of the result of Step 5. The result is the standard deviation. x (x i x) (x i x) x Variance: s = (x x) n 1 = = 5.7 Standard Deviation: s = 5.7 =.387.4

6 Shortcut formula Sample Variance s = n (x ) ( ) n(n 1) Sample Standard Deviation n (x ) ( ) s = n(n 1) Round-Off Rule for Measures of Variation When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data. Round only the final answer, not values in the middle of a calculation. Properties of the Standard Deviation (1) 1. n 1 is called the degrees of freedom.. s measures variability about the mean and should be used only when the mean is chosen as the measure of center. 3. s is always zero or greater than zero. s = 0 only when there is no variability. This happens only when all observations have the same value. Otherwise, s > As the observations become more variable about their mean, s gets larger. 5. s has the same units of measurement as the original observations. For example, if you measure weight in kilograms, both the mean x and the standard deviation s are also in kilograms. This is one reason to prefer s to the variance s, which would be in squared kilograms. 6. Like the mean x, s is not resistant. A few outliers can make s very large. Properties of the Standard Deviation () The standard deviation measures the variation among data values. The standard deviation is a measure of variation of all values from the mean. Data values close together A small standard deviation Data values with much more A larger standard deviation Ex. 4., 3.5, 3., 4.0, 4.1 S.D : Ex. 5, 7, 1,, 4 S.D :

7 Spotting suspected outliers (Mean as the center) Using a mean and the standard deviation to analyze data. Range Rule of Thumb: The vast majority (such as 95%) of sample values lie within two standard deviations of the mean for many data set. Minimum usual value = mean standard deviation Maximum usual value = mean + standard deviation