Measures of Central Tendency: Averages or other measures of “location” that find a single number that reflects the middle of the distribution of scores— ”the average score”

You know it better as the “average” Total/n = mean = 48/6 = 8 cups per person How many cups of coffee (x) Joe12 James10 Jane10 Chris8 Fred5 Christina3 TOTAL (n=6 students surveyed)48

Summation Symbol: Σ Σx: sum of x Σx/n = mean = 48/6 = 8 cups per person How many cups of coffee (x) Joe12 James10 Jane10 Chris8 Fred5 Christina3 TOTAL (n=6 students surveyed)48

Σx/n = x-bar (a.k.a ) = mean Σx/n = mean = 48/6 = 8 cups per person How many cups of coffee (x) Joe12 James10 Jane10 Chris6 Fred5 Christina5 TOTAL (n=6 students surveyed)48

The ungrouped frequency distribution for our raw collected data: BEWARE!! Averaging number of cups is not as simple with frequency distributions… How many cups of coffee (x)f = n =6

The ungrouped frequency distribution for our raw collected data: For ungrouped frequency distributions, Σfx/n = Σfx/Σf = 48/6 = 8 How many cups of coffee (x)f =fx = = n = Σf48 = Σfx

Checking your work: Σ(x – xbar) = 0 How many cups of coffee (x) xbarx-xbar Joe1284 James1082 Jane1082 Chris68-2 Fred58-3 Christina58-3 TOTAL (n=6 students surveyed) 48

A value in which there are as many scores greater than the median as there are scores less than the median First, order the raw scores from highest to lowest Second, find the median position (or median subject) by using this formula: median position = (n + 1)/2 Third, find the median score associated with that position What is the median value of these raw scores: 12, 12, 10, 10, 6, 5, 5

median position = (n + 1)/2 = (7 + 1)/2 = 4  If done properly, should be able to count from bottom or top and get same value How many cups of coffee (x) Joe12 Jessica12 James10 Jane10 Chris6 Fred5 Christina5

median position = (n + 1)/2 = (7 + 1)/2 = 4 th person (or i=4)  Median value = value of x at median position Person Interviewed or Studied (i) How many cups of coffee (x)

Using our original example, we have 6 people studied, which leads to an added complication in our median calculation median position = (n + 1)/2 = (6 + 1)/2 = 3.5 th person (or i=3.5) In this case, you take the value at 3 rd position and the value at the 4 th position and take the midpoint between: i=3 gives you 10 and i=4 gives you 8, midpoint would be 9 Person Interviewed or Studied (i) How many cups of coffee (x)

The mean is influenced by extreme values while the median is not The median is still 9 in this example, but the mean would be more than two trillion! Person Interviewed or Studied (i) How many cups of coffee (x) 112,000,000,000,

The grouped frequency distribution for our raw collected data:  Remember to find the median position: = (n + 1)/2 We know that for grouped frequency distributions, n = Σf Therefore, (n + 1)/2 = (Σf + 1)/2 = (9 + 1)/2 = 5 How many cups of coffee (x)f = = n = Σf

Mean of frequency data: Σfx/n = Σfx/Σf = 62/17 = 3.65 Possible # of cups of coffee (x) # of people who had that many cups (f) # of cups (fx)Cumulative frequency (cf) Σf = 17Σfx = 62

Median position of frequency data: (n+1)/2 = (Σf + 1)/2 = 18/2 = 9. The subject with median of 9 enters when x=4. Median = 4 Possible # of cups of coffee (x) # of people who had that many cups (f) # of cups (fx)Cumulative frequency (cf) n = Σf = 17Σfx = 62 

The mode: a category of a variable that contains more cases than either of the adjacent categories The mode is not influenced by extreme values (just like the median), but modal categories may disappear as sample size increases. Possible # of cups of coffee (x) # of people who had that many cups (f)  

Measures of Central Tendency: Averages or other measures of “location” that find a single number that reflects the middle of the distribution of scores— ”the average score”—mean, median, mode Measures of Dispersion: measures concerning the degree that the scores under study are dispersed or spread around the mean—”variability” Range, mean deviation, variance, standard deviation

As we’ve discussed with the means and medians, there are two different approaches to calculating the dispersion (i.e., variability around a mean) for raw scores and frequency data!

The range compares highest score and the lowest score for a given set of scores. It is the simplest of all dispersion measures. Example: We calculated an average of 8 cups of coffee per student with our survey of 48 students. The reported number of cups ranged from 0 to 12. Not terribly useful because the measure is HEAVILY influenced by extreme values without regard to all other numbers.

Just as the range only measures the extreme ends and ignores all other values in the data, the mean deviation is actually sensitive to all values in the dataset. Essentially, it measures 1) how different each value in a dataset deviates from the mean, 2) sums up all these differences across all observed values to get the total amount of deviation, 3) and dividing this sum of deviations by the total number of scores in dataset to get an average deviation. The mean deviation: an average distance that a score deviates from mean

Σx/n = x-bar ( ) = mean Σx/n = mean = 48/6 = 8 cups per person How many cups of coffee (x) x - Joe1284 James1082 Jane1082 Chris68-2 Fred58-3 Christina58-3 TOTAL (n=6 students surveyed) 48

MD=( )/6 = 16/6 = 2.67 MD= Σ |x- | n How many cups of coffee (x) x - Joe1284 James1082 Jane1082 Chris68-2 Fred58-3 Christina58-3 TOTAL (n=6 students surveyed) 48

The variance (s 2 ) is an “average” or mean value of the squared deviations of the scores from the mean Computationally, the equation is very similar to the mean deviation except instead of absolute values the variance considers the true values and squares them Variance= s 2 = Σ (x- ) 2 n The variance is usually much larger than the mean deviation because we are taking the squares of the deviations MD= Σ |x- | n

Variance= s 2 = Σ (x- ) 2 = 46/6 = 7.67 n How many cups of coffee (x) x -(x- ) 2 Joe James10824 Jane10824 Chris68-24 Fred58-39 Christina58-39 TOTAL (n=6 students surveyed) 4846

Since the variance (s 2 ) is usually larger than the mean deviation because we take the squares of the deviations in the calculations, another method for calculating the variability around a mean is used to “standardize” the variance—the standard deviation (s) Essentially, the standard deviation is the same as the variance except we take the square root of the variance estimate to calculate it: Standard Deviation= s = This estimate aims to provide a measure of dispersion closer in size to the mean deviation than the variance MD= Σ |x- | n

Standard deviation= s = square root of (Σ (x- ) 2 n 7.67 = s 2 s = square root of 7.67 = 2.77 How many cups of coffee (x) x -(x- ) 2 Joe James10824 Jane10824 Chris68-24 Fred58-39 Christina58-39 TOTAL (n=6 students surveyed) 4846

Remember for raw scores, we calculated the variance as: (raw scores) Variance= For frequency data, the equation is adjusted so that we multiply each squared deviation by the frequency of that particular value of x: (frequency data) Variance = The standard deviation is simply the square root of that variance MD= Σ |x- | n

How many cups of coffee (x) f =fx = x- (x- ) 2 (x- ) 2 X f = n = Σf 48 = Σfx 46 (frequency data) Variance = = s 2 = 46/6 = 7.67 (frequency data) St. Dev = s= square root of 7.67 = 2.77