Download Analyzing Data: Measuring Center & Variability with Mean, Median, Mode, Range, Variance, & and more Study notes Statistics in PDF only on Docsity! 27 STAT 100A Chapter 2: Describing Data with Numerical Measures Sections 2.1-2.2: Measures of Center Measures of Center 1. mean (average) โ center of gravity 2. median (middle) โ separates the bottom 50% of the data from the top 50% of the data 3. mode (most) โ most frequent observation Note: We will represent data values as x1, x2, x3, โฆ, xn. Example: For the data set 10, 7, 5, 4, 11, 13, 8, and 3, identify the xiโs. 28 Summations ๏ฑ๏ฎ๏ ๏x = x1 + x2 + x3 + โฆ + xn ๏ ๏ ๏ฒ๏ฎ๏ ๏x2 = x12 + x22 + x32 + โฆ + xn2 3. (๏x)2 = (x1 + x2 + x3 + โฆ + xn)2 ๏ ๏ ๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏๏ ๏ Example: Find the 3 summations above for the following data set. 10 7 5 4 11 13 8 3 29 sample mean - the sum of all of the values in a data set divided by the number of observations. sample median - the value that lies in the middle of a data set when the data are arranged from low to high. The position of the median is (n+1)/2. Steps in Computing the Median of a Data Set 1. Arrange the data from low to high. 2a. If n is odd, the median is the unique middle data value. 2b. If n is even, the median is the mean of the two middle observations in the data set. sample mode - the most frequent observation that occurs in the data set. bimodal - multimodal โ modal class โ Which measure of center could also be used with qualitative data? 30 Cell Phone Usage The following data represent the monthly cell phone bill for one person for six randomly selected months. $35.34 $42.09 $39.43 $38.93 $43.39 $49.26 Compute the mean, median, and mode phone bill. 31 Flight Time The following data represent the flight time (in minutes) of a random sample of seven flights from Las Vegas, NV, to Newark, NJ, on Continental Airlines. 282 270 260 266 257 260 267 Compute the mean, median, and mode flight time. 32 The Shape of the Distribution and the Mean and the Median 33 outlier โ an observation that falls well above or well below the overall bulk of the data. ๏ท The median is more resistant to an outlier than the mean is. ๏ท If a distribution is very highly skewed, the median is usually preferred over the mean because it better represents what is typical. ๏ท If the distribution is close to symmetric or only mildly skewed or if it is discrete with few distinct values, the mean is usually preferred because it uses the numerical values of all the observations. 34 STAT 100A Section 2.3: Measures of Variability Measures of Variability sample range - the difference between the largest data value and the smallest data value. range = H - L sample variance - the average squared deviation about the sample mean. s2 = OR s2 = Note: Why n-1? If we divided by n, the sample variance would consistently underestimate the population variance. Whenever a statistic consistently underestimates or overestimates a parameter, it is called a biased estimate. To obtain an unbiased estimate of the population variance, we divide by n-1.
