   ## Statistics I Lab 1, Fall 2012 -- "Middles"

 Objectives This page does two things: reviews the descriptive statistics from middle school, introduces the descriptive statistics used in college.   Definitions and answers are at the bottom of the page.

 Medians     The median is the middle number when the numbers are ordered, listed, from lowest to highest. Find the median.1. Sample: 2, 4     answer 2. Sample: 2, 2     answer 3. Sample: 1, 5     answer 4. Sample: 5, 1     answer 5. Sample: 3, 7, 2     answer 6. Sample: 0, 4, 0, 4     answer 7. Sample: 1, 3, 5, 4, 2, 6, 5     answer 8. Sample: 1, 3, 5, 4, 3, 2, 6, 5     answer Modes     The mode is the most frequent data point. The score that has the highest frequency, the one that occurs the most. Find the mode.9. Sample: 3, 3, 5, 7     answer 10. Sample: 1, 2, 3, 5, 5, 7, 7, 6, 4     answer 11. Sample: 3, 4, 3, 4, 6, 9, 5, 5, 6, 9     answer Mean     The mean is the arithmetic average, the sum of the data points divided by the number of data points.     This formula is often used: .   More Info Find the mean. 12. Sample: 2, 4, 5, 7     answer 13. Sample: 1, 2, 3     answer 14. Sample: test grades of: 100% , 90%, 50%, 75%, 80%     answer 15. answer Midrange     The midrange is the mean of the highest and lowest data points. It is the [(highest score) + (lowest score)]/2. Find the midrange. 16. Sample: 2, 4, 5,     answer 17. answer Report on the Sample     The last problem on the page is to write a short report on the statistics of the following sample. For as many of the terms listed in the vocabulary list, name the statistic and give its numerical value. That's it.     18. Sample: 1, 2, 4, 7, 5, 9, 20, 12, 12.     answer Vocabulary   AVERAGE     - a centrally positioned number . The position may be cased on frequency, or on rank, or on a formula. The average many not even be included in the original sample or population, but, it is centrally positioned based on some definition. Very often the mean is considered the average, but, it is only one of many possilbe averages.   More Info CENTER     - any one of the following measures of the middle of the distribution of data: the mean, the median (also 2nd quartile, also 50th percentile), the mode, the midrange.   DATA or DATA POINT, x     - a piece of data; a number in the sample of data, a number in the population of data.   50th PERCENTILE     - the median, the 2nd quartile, the middle number when the numbers are ordered from lowest to highest. The 50th percentile may not be a member of the sample or the population but the generated using the mean of the two middle numbers when the sample has an even number of pieces of data.   FREQUENCY f(x)     - the number of times a piece of data is found in the sample or population.     More Info PARAMETER     - a number used to describe or name a population or sample.   POPULATION     - the entire set of data.   MEAN, - the arithmetic average - the sum of the data points divided by the number of data points More Info MEDIAN, - 50th percentile - the 2nd quartile, the middle number when the numbers are ranked from lowest to highest. The 50th percentile may not be a member of the sample or the population but the generated using the mean of the two middle numbers when the sample has an even number of pieces of data.     If you have an odd number of data points the middle number is easy to find. It is in the ordered middle.     If you have an even number of data points the two middle numbers are averaged, their mean is computed and used as the median.   More Info MIDRANGE     - [(greatest data point) + (smallest data point)]/2     ,   [hi + lo] / 2   MODE     - most frequent data point, data point with the highest frequency.   More Info 2nd QUARTILE, Q2     - median - the 50th percentile, middle number when the numbers are ordered from lowest to highest. It may not be a member of the sample or the population but the generated once all data points have been ordered.   SAMPLE     - a subset of a population   SAMPLE SIZE, n     - a number of data points in a sample.

 Answers 1. 3: 3 is the median, , of 2 and 4. It exactly in the middle of 2 and 4.  2. 2: there is no other number between the two 2s   3. 3: 3 is the median, , of 1 and 5. It exactly in the middle of 1 and 5. Notice that 3 is the median of 1-2-3-4-5. Three is the mean of 1 and 5, the mean must be used because there are only 2 numbers. (1+5)/2 = 3   4. 3: order the numbers 1, 5 1*3*5, 3 is in the middle, the median position; (1+5)/2 = 3:   the median, , is 3, is also the mean, , of 1 and 5  5. 3: because 3 , 7, 2 is 3 numbers the median is easy to find. Order the numbers, 2, 3, 7 , is 3  6. 2: 0, 4, 0, 4, 0,0,4,4, is the mean of the middle 0 and 4 is 2  7. 4: 1, 3, 5, 4, 2, 6, 5 1, 2, 3, 4, 5, 5, 6. There are an odd number of data points, the median is visible in the ordered middle position. The median is 4.  8. 3.5: 1, 3, 5, 4, 3, 2, 6, 5 1, 2, 3, 3, 4, 5, 5, 6, the median is 3.5 the mean of 3 and 4   9. 3: 3 occurs most  10. 5, 7: 5 and 7 have the highest and equal frequencies, they are each considered modes. It is a special case. The sample is said to be bimodal, having 2 modes.  11. There is no mode because each data point occurs with the same frequency.  12. 4.5: (2+4+5+7)/4 = 18/4 = 4.5  13. 2: (1+2+3)/3 = 6/3 =2. Notice, the mean is also the median.  14. 79%: (100+90+50+75+80)/5 = 395/5 = 79%  15. 16. 3.5: 2 is the lo. 5 is the hi. (2+5)/2 = 3.5 Notice it is close to the median which is 4 and the mean which is 3 2/3.  17. 18. There are 9 data points in the sample. The mean is 8, the mode is 12, the median is 7, and the midrange is 10.5. The scores range from a low of 1 to a high of 20. The mode has a frequency of 2.One might also add: "The data points are: 1, 2, 4, 5, 7, 9, 12, 12, and 20," but this is not usually included.      © 2012 A. Azzolinowww.mathnstuff.com/math/spoken/here/2class/90/fclab1.htm