3.1 Measures of Central Tendency: Mode, Median and Mean. Average a single number that is used to describe the entire sample or population

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1 . Measures of Central Tendency: Mode, Median and Mean Average a single number that is used to describe the entire sample or population. Mode a. Easiest to compute, but not too stable i. Changing just one number in a data set can change the mode dramatically ii. Useful when we want to know the most frequently occurring data value, such as the most frequently requested shoe size. b. Used with all four levels of data (nominal, ordinal, interval or ratio c. The value or property that occurs most frequently in the data set d. Not every data set has a mode i. E. If Professor Fair gives equal numbers of As, Bs, Cs, Ds, and Fs, then there is no modal grade.. Median a. The central value of an ordered distribution. To find it: i. Order the data from smallest to largest. ii. For an odd number of data values in the distribution, Median = Middle data value iii. For an even number of data values in the distribution, Sum of middle two values Median = b. More stable than the mode c. Can be used with the three higher levels of data (ordinal, interval and ratio d. The median uses the position of the number, rather than a specific value of each data entry i. If the etreme values of a data set change, the median usually does not change (The is why the median is often used as the average for house prices. e. E. What do BBQ flavored potato chips cost? The price per ounce in cents of the rated chips are: Mean a. Add the values of all the entries and then divide by the number of entries b. Uses the eact value of each entry c. Most stable way to compute the average d. Can be used with the two highest levels of data (interval, ratio

2 Notation The symbol for the mean of a sample distribution of values is denoted by (read as bar. If your data comprises the entire population, we used the symbol µ (lowercase Greek letter mu to represent the mean. The procedure to compute the mean is the same regardless of whether we have population or sample data. If we let n represent the number of entries in a sample data set and N represent the number of entries in a population data set, the formulas are: Sample mean Σ Σ = = Population mean = µ = n N Σ is read as the summation of. represents each individual entry and Σ means that you will add all of the individual entries together. Resistant measures A resistant measure is one that is not influenced by etremely high or low data values. The mean is not a resistant measure of center because we can make the mean as large as we want by increasing the size of only one data value. The median, on the other hand, is more resistant. However, a disadvantage of the median is that it is not sensitive to the specific size of a data value. Trimmed Mean A measure of center that is more resistant that the mean but still sensitive to specific data values is the trimmed mean. A trimmed mean is the mean of the data values left after trimming a specific percentage of the smallest and largest data values from the data set. Usually a 5% trimmed mean is used. This implies that we trim the lowest 5% of the data as well as the highest 5% of the data. To compute a 5% trimmed mean:. Order the data from smallest to largest. Multiply.05 times the number of data entries (If this does not produce a whole number, then round to the nearest integer. Delete that number (from step of data entries from the bottom and top of the data set. 4. Compute the mean of the remaining 90% of the data. EXAMPLE A sample of 0 colleges in California showed class size for introductory lecture courses to be: Compute the mode, median and mean. Then compute a 5% trimmed median and a 5% trimmed mean. Did the trimmed values change? Is the trimmed mean or the original mean closer to the median?

3 . Measures of Variation An average is an attempt to summarize a set of data in just one number, but sometimes this isn t enough. We need a statistical cross-reference that measures the spread of the data. Range the difference between the largest and smallest values of a data distribution Standard deviation a value that tells how the data entries differ from the mean The formula for the standard deviation differs slightly depending on whether you are using an entire population or just a sample. For now, we will compute the standard deviation for sample data only. Σ( Sample standard deviation = s = pop. Stan.dev. = σ = Σ ( µ N any entry in the distribution the sample mean n the number of entries in the sample µ the population mean N the number of entries in the population σ population standard deviation (Lowercase greek letter, pronounced sigma To compute this formula:. List each data entry ( L. Subtract the mean ( from each data entry ( (. Square the values from step ( or 4. Add the values from step ( Σ( 5. Divide the value of step 4 by (n- L = (L Σ or sum L ( or or L L sum( L Σ( 6. Take the square root of the value in step 5 or = sum( L Sample Variance a measure of the spread or dispersion within a set of sample data. The variance is the step before the square root of the above equation. Σ( Sample variance = = sum( L s or

4 The value Σ ( is used so commonly in other measures of statistics that it has its own name, the sum of squares, and symbol computation of that epression is: SS. A more simple Sum of squares ( Σ = SS = Σ( = Σ n To compute: L = values L ( = L Then SS s = Find the sum of L and L ( sum( L SS = sum( L n Why we divide by n- and N: The reason is that N represents the population size, while n represents the sample size. Since a random sample usually will not contain etreme data values (large or small, we divide by n- in the formula for s to make s a little larger than it would have been had we divided by n. Courses in advanced theoretical statistics show that this procedure will give us the best possible estimate for the standard deviationσ. If we have the population of all data values, then etreme data values are present, so we divide by N instead of n-. Coefficient of Variation A disadvantage of the standard deviation as a comparative measure of variation is that it depends on the units of measurement. This means that it is difficult to use the standard deviation to compare measurements from different populations. For this reason, statisticians have defined the coefficient of variation, which epresses the standard deviation as a percentage of the sample or population mean. If and s represent the sample mean and sample standard deviation, then the sample coefficient of variation CV is defined to be: CV s = 00 Chebyshev s Theorem For any set of data: At least 75% of the data fall in the interval from µ σ to µ + σ At least 88.9% of the data fall in the interval from µ σ to µ + σ At least 9.8% of the data fall in the interval from µ 4 σ to µ + 4σ

5 . Mean and Standard Deviation of Grouped Data If you have a lot of data values it can be quite tedious to compute the mean and standard deviation. In many cases, a close approimation to the mean and standard deviation is all that is needed, and it is not difficult to approimate these two values from a frequency distribution. Treat each entry of a class as though it falls on the midpoint ( of that class. Then the midpoint times the number of entries in a class ( f represents the sum of the observations in the class. The formulas for the mean of grouped data and standard deviation of grouped data are as follows: Sample mean for a frequency distribution Σf = Sample standard deviation for a frequency distribution s = ( Σ f To find the standard deviation of grouped data you will need 7 columns. I Class : add the two numbers of the classes and divide by II Midpoint ( III Frequency ( f IV Multiply column II times column III ( f Then add columns II and IV and divide the sum of column IV by the sum of Σf column II and this will give you the mean = V Subtract each the mean from each value in column III VI Square each value from column V ( VII Multiply column VI times column II ( f Then add column VII and divide the sum by the sum of column II minus Σ( variance: s = f Σ( Then take the square root of the answer, standard deviation s = f

6 Shortcut formula Computation formula for the standard deviation SS s = Where SS = Σ f ( Σf is the midpoint of a class, f is the number of entries in that class, Σ f is the total number of entries in the distribution The summation Σ is over all classes in the distribution. To compute this formula:. List your values (midpoints in column (. List the corresponding frequencies on column ( f. In column, multiply column times column (f 4. In column 4, square column and then multiply it times column ( f 5. FIND THE MEAN a. Add column (f and column ( f Σf b. Divide the sum of column by the sum of column = 6. FIND THE VARIANCE AND STANDARD DEVIATION ( Σf a. SS = Σ f n b. You will take the sum of column and square it and then divide by n (which is actually Σ f c. Subtract what you get in part b by the sum of column 4. This is SS SS d. The variance is s = Σ f e. The standard deviation is = SS s Weighted Average Σw Σw. Take each weight (w and multiply by its corresponding score (.. Add to get Σ w.. Add the weights to get Σ w. 4. Divide the answer for step by the answer for step.

7 .4 Percentiles and Bo and Whisker Plots th For whole number P (where P 99, the P percentile of a distribution is a value such that P% of the data fall at or below it and (00-P% of the data fall at or above it. There are 99 percentiles, and in an ideal situation, the 99 percentiles divide the data set into 00 equal parts. Since the operative word is ideal it seldom happens. Many different methods are available to determine the eact value of a percentile. The only percentiles that we will worry about are the quartiles. Quartiles the percentiles that divide the data into fourths. The first quartile, Q, is the 5 th percentile, the second quartile, Q, is the median, the third quartile, Q, is the 75 th percentile. Procedure to compute quartiles. Order the data from smallest to largest. Find the median. This is the nd quartile. (An easy way to find the median n + is, where n is the number of items.. The first quartile, Q, is then, the median of the lower half of the data; that is, it is the median of the data falling below the Q position (and not including Q. 4. The third quartile, Q, is then the median of the upper half of the data; that is, it is the median of the data falling above the Q position (and not including Q. The median, or second quartile, is a popular measure of the center utilizing relative position. A useful measure of data spread utilizing relative position is the interquartile range (IQR. It is simply the difference between the third and first quartiles. Interquartile range = Q Q

8 Bo and Whisker Plots The quartiles together with the low and high data values give us a very useful five-number summary of the data and their spread. Five-number summary Lowest value, Q, median, Q, highest value To make a bo-and-whisker plot. Draw a horizontal scale to include the lowest and highest data values.. Above the scale you will put dots at the points from your 5 number summary.. Connect the dots from the highest value to the rd quartile and the lowest value to the st quartile. 4. Draw a bo from the rd quartile to the st quartile so that the lines drawn from the highest value to the rd quartile and the lowest value to the st quartile are centered on the bo. 5. Draw a solid line through the bo at the median level. Eample (pg. 8 Low value = ; Q = 8, median =.5; Q = 9; high value =