Math 155. Measures of Central Tendency Section 3.1

Transcription

1 Math 155. Measures of Central Tendency Section 3.1 The word average can be used in a variety of contexts: for example, your average score on assignments or the average house price in Riverside. This is where one number is used to describe the entire sample or population. This section will look at three different ways to describe central tendencies: mode, median and mean. Mode. The mode of a data set is the value that occurs most frequently. Example 1. Find the mode of the data For this data, the mode is 4, as it occurs 4 times. Note that the numbers were just the number of letters in the sentence describing the mode. For a large data set, it would be useful to sort or order the numbers to find the mode. It is possible a data set has no mode, for example, if all numbers occur equally often. Sometimes we will say a data set is bimodal if there are two distinct number that occur most often. Median. The median is the central value of an ordered distribution of data. Procedure to find the median 1. Order the data from smallest to largest. For an odd number of data values in the distribution, Median = Middle data value 3. For an even number of data values in the distribution, Median = Sum of middle two values Example. Find the median of the following simple data sets. Observe the difference when there is an odd number or even number of data. (a) (b) Solution. (a) This set of data has an odd number of data, so the median is the middle value The highlighted middle value is 5, and so the median is 5. (b) This set of data has an even number of data, so the median is the average of the middle two values So the median is average of 5 and 11 is = 8 (observe 8 is right in the middle of 5 and 11).

2 Example. Consider the following sample consisting of 0 numbers. (a) Find the mode of the data (b) Find the median of the data Solution: (a) The mode is 39, the most frequently occurring number in the data set, as highlighed below (b) Because there is an even number of data, the median is the average of the middle two values in the ordered data, and so the median is = 35.5 Example 3. Consider the following data set consisting of 19 numbers. (a) Find the mode of the data (b) Find the median of the data Solution: (a) The mode is 0, the most frequently occurring number in the data set as is highlighted below (b) The median is in the central position (10th place) of the ordered data, so the median is 34 as highlighted below Position of the median. For small sets of ordered data, it is fairly easy to scan the data and identify the middle, or the middle two pieces of data. For larger sets, it is simpler to identify the median by identifying its position as follows. For an ordered data set of size n, Position of the middle value = n + 1

3 This says the position of the median in the data set of size n = 0 given in Example is in the = 10.5th position. This means take the average of the 10th and 11th data as was done. This says the position of the median in the data set of size n = 19 given in Example 3 is in the = 10th position, as was done above. Example 4. (a) Given a collection of ordered data with 3 numbers, in what position is the median? (b) Suppose the data in positions 159 to 165 for an ordered data set of size 3 is as below. Find the median of the data set (That is 50 is in position 159, 54 is in position 160, etc.) (c) Given a collection of ordered data with 53 numbers, in what position is the median? (d) Suppose the data in positions 15 to 131 for an ordered data set of size 53 is as below. Find the median of the data set (That is 50 is in position 15, 54 is in position 16, etc.) Mean. An average that uses that uses the exact value of each entry is the mean, and it is computed as follows Sum of all entries x Mean = Number of entries = when n = number of entries n The mean is the average teachers usually use to compute grades. When the data is based on a sample, we use x to denote the sample mean statistic. When the data is from the entire population, we use µ to denote the population mean parameter.

4 Example 5. A student received the following grades in their chemistry tests last quarter (there was a test every Friday except for during the first and last weeks of the quarter) (a) Find the mode. (b) Find the median (c) Find the mean score. (Note: the sum of the test scores is 594). (d) In actuality in that class, the instructor dropped the highest and lowest test scores and then computed the mean of the remaining 6 tests (this is called a trimmed mean). Compute this trimmed mean for the test scores. What was the median of the remaining 6 test scores? Round answers to one decimal place, when needed in the above answers. A common trimmed mean that is used is a 5% trimmed mean. That means the highest and lowest 5% of the data are deleted and the mean is computed on the remaining 90% of the data. See your worksheet for further examples. Weighted averages are common in computing things such as grades or GPA s where different categories have different weights. In the GPA example, the weights are the number of units per course. The formula for a weighted average is Weighted Average = xw w where x is the value and w is the weight. Example 6. Suppose a student took 10 units of courses and had an A in a 3 unit course, a B in a 4 unit course, a C in a 1 unit course and an F in a unit course. The we would compute the weighted average (GPA) as follows: x w xw Sums: 10 4 Thus w = 10 and xw = 4 and so the GPA is the weighted average = 4 10 =.4. Think about how this would compare if we made each grade (in its numeric equivalent) at the frequency of the course units and took the mean, that is if we computed the mean of 4, 4, 4, 3, 3, 3, 3,, 0, 0...

5 An interesting example is as follows Example 6. On-time percentages are given for two airlines in Phoenix, Los Angeles and Seattle for last year. Crashcade Airlines Los Angeles Phoenix Seattle Number of Fights On time % Pacific Worst Airlines Los Angeles Phoenix Seattle Number of Fights On time % (a) Calculate the on-time percentage average for these three cities for each airline. weighted average where the weight for each airline and city is the number of flights. Do this as a (b) Given that the on-time percentage for Crashcade Airlines is 5% higher in each city, does the answer in (a) surprise you? Why or why not? Solution: (a) For Crashcade we compute xw (1000)(90%) + (500)(95%) + (3500)(85%) 435, 000 = = = 87% w For Pacific Worst we compute xw (50)(85%) + (4500)(90%) + (50)(80%) 446, 50 = = = 89.5% w You should get the same answer if you computed this as follows (we illustrate with Crashcade): Out of Los Angeles, 90% of 1000 flights = 900 flights were on time, out of Phoenix, 95% of 500 = 475 flights were on-time, out of Seattle 85% of 3500 = 975 flights were on time. Therefore, Crashcade had a total of = 4350 out of 5000 flights on time, which is 87%. (b) On the surface it is very surprising that Pacific Worst has a better overall on-time percentage. However, this happens because Pacific Worst s schedule is heavily weighted to flights in Phoenix where they have their best on-time percentage, whereas Crashcade s flights heavily weighted in Seattle where they have their worst on-time percentage.