We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:


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2 MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having the greatest frequency if the class intervals are equal and the mode is the midpoint of the modal class. Example: Obtain the mode for Data Set 2 Modal interval is For a more exact number, we say that the mode is the midpoint of the interval or 74.5 Properties of Mode 1. Appropriate for all types of data. 2. Not necessarily unique. 3. Note that the mode doesn't have to be near the center of the distribution. 2
3 MEDIAN The median of a set of observations is the value of the variable such that half the values are less than the median and half are greater than the median. When observations are ordered from lowest to highest, the median is the number that divides the sample so that an equal number of cases fall above and below it. For discrete observations, the median is found by first ordering the observations from smallest to largest, and then if the number of observations, n, is odd, taking the middle observation (n+1)/2 and if n is even, the median is the average of the observations at position n/2 and (n/2)+1 in the ordered arrangement. Example: Obtain the median for Data Set 1 10th and 11th scores are 77 and 71. Average of these two scores is 74. For a grouped frequency distribution, the median lies with the class (the median class) containing the value with a cumulative frequency of (n+1)/2. It can be determined from a cumulative frequency polygon or numerically by interpolation within the median class. Example: Obtain the median for Data Set 2 We're looking for the interval that contains the middle observation. Looking at cumulative frequency numbers, that would be interval (critical interval). Whereabouts in the interval is the median or that score in the 50th percentile? Md = l i + i [((n+1)/2) f m /f i )] Md = median l i = exact lower limit of interval i i = width of interval i (n+1)/2 = identifies interval in which median is located f m = number of observations in intervals below interval i f i = number of observations in interval i Median: [(10.5 7)/6]*10 = Properties of Median 1. Appropriate for interval level data and ordinal data. 2. Median is not affected by extreme scores. 3. Median is insensitive to distances of measurements from the middle. 4. The median is problematic with binary data. Quartiles and other Percentiles Percentile: pth percentile is a number such that p% of the scores fall below it and (100p)% fall above it. Lower quartile: 25th percentile  median for observations that fall below median Upper quartile: 75th percentile  median for observations that fall above median 3
4 MEAN The sample mean of a sample of n observations x 1, x 2,, x n, denoted by x, is the sum of the n observations divided by n if the data are not grouped. x = sum of all observations in the sample number of observations in the sample = (x 1 + x x n )/ n = (Σ x) / n where n is the number of observations, x 1 is the first sample observation, x 2 is the second sample observation,, x n is the nth (last) sample observation. The population mean, denoted by µ, is the average of all x values in the entire population. Example: Obtain the sample mean for Data Set 1 Sum up all scores and divide by 20: 1466/20 = 73.3 Weighted Mean x = (n 1 x 1 + n 2 x 2 ) / (n 1 + n 2 ) Example: Class 1: n 1 =20; x 1 = 73.3 Class 2: n 2 =15; x 2 = 81 [20(73.3) + 15(81)] / [20+15] = 76.6 Mean for Grouped Frequency Distribution Example: Obtain the mean for Data Set 2 Weight midpoints of class intervals by the frequency of observations in each interval. [3(94.5)+4(84.5)+6(74.5)+3(64.5)+2(54.5)+2(44.5)] / 20 = 73 Properties of Mean 1. Appropriate only for interval level data and above. 2. Mean is very sensitive to extreme scores, called outliers. An additional measurement at outer points would pull up or down the mean. So mean may not be representative of the measurements in the sample (particularly with small samples). 3. Mean is the center of gravity or point of balance for frequency distribution. 4. The sum of the deviations from the mean equal 0. 4
5 The deviation indicates the distance and direction of any raw score from the mean. Deviation = X  x Sum of the deviations = Σ (X  x ) = 0 Example 1, 3, 4,4, 9, 15 x = 6 Deviations: X  x (16)=5 (36)=3 (46)=2 (46)=2 (96)=3 (156)=9 Σ (X  x ) = 0 5
6 Shape of Distribution Mean, mode and median are identical for a unimodal, symmetric distribution, such as bellshaped distribution. The mean and median are identical if the histogram is symmetric. (e.g. Bimodal distribution) If the histogram is unimodal with a long right hand tail (positively skewed) the mean lies above the median. 1, 2, 3, 4, 100 Mean = 22; Median = 3 If the histogram is unimodal with a long left hand tail (negatively skewed), then the mean is smaller than the median. 1, 97, 98, 99, 100 Mean = 79 Median = 98 Mean is influenced by extreme scores so you should be cautious about using the mean with highly skewed distribution. Median is not affected much, if at all, by changes in extreme scores. With a bimodal distribution, it's best to characterize the distribution by both modes. Using median or mean would obscure important features of the distribution. In summary, measures of center (also known as measures of location) describe the center of a data set, or the location of a typical value, using a single value. There is no best measure of location, and the different measures of location do not measure the same thing. Each measure of location summarizes only one aspect of a data set and they should not be looked at in isolation. 6
7 Describing Variability in a Data Set Besides wanting to know the center of the data, we are interested in how far individual values in the sample are from this center. A measure of variability (or measure of dispersion) is used to describe how the data are spread about its center. A small value indicates that the data are grouped closely together while a large value indicates that the data are spread out. Measures of variability can be used to compare several distributions. To illustrate measures of dispersion, we ll use the following two data sets. The information in these data sets represents test scores for individuals in each of the two classes. CLASS Mode=6; Median=6.5; Mean=7 CLASS Median=13.5; Mean=13 We'll consider three different measures of dispersion: Range ( R) Variance (Sample = s 2 ; Population = σ 2 ) Standard Deviation (Sample = s; Population = σ) 7
8 Range (R) Provides a quick but rough measure of variability. Difference between the minimum (low L) score and maximum (high H) score. R=HL Example: Class 1: 10 5 = 5 Class 2: 20 4 = 16 Properties of range 1. Sensitive to sample size 2. Unstable Interquartile Range A measure of variation based on the quartiles of a distribution. IQR = Q 3 Q 1. Q 3 refers to the upper quartile or the score in the 75 th percentile. Q 1 refers to the lower quartile or the score in the 25 th percentile. Semiinterquartile range = (Q 3 Q 1 ) / 2 = average of the difference between the third and first quartiles. Properties of IQR 1. Not sensitive to extreme outlying observations, unlike the ordinary range. 2. IQR increases as variability increases. 3. IQR can mask real differences in a distribution. Other measures of variation are based on the deviations of the data from a measure of central tendency, usually their mean. Review of summation notation Suppose c is some constant 1. Σ cx i = cx 1 + cx cx n = c(x 1 + x x n ) = cσ x i 2. Σ (x i + y i ) = Σ x i + Σy i 3. Σ c = c + c + c c = nc 4. Σ (c + dx i ) = Σ c + Σdx i = nc + dσx i 5. Σ x i + c Σ (x i + c) 8
9 Recall that the mean is considered to be the center of gravity for a set of observations. Σ (X  x ) = 0 Deviation of ith observation X i from the sample mean x is (x i  x ), the difference between them. Deviation + when an observation is greater than sample mean. Deviation  when an observation is less then sample mean. To obtain mean deviation, we would sum up all the individual deviations from the mean and divide by n, the sample size. 1/n Σ (x i  x ) = 0 Example Class 1 Class = = = = = = = = = = = = 9 Because mean is center of gravity, the sum of the positive deviations equals the sum of the negative deviations and sum of all deviations about mean is 0. There are several ways we can avoid the sum of deviations equaling 0. One is to take the absolute value of the deviations. Mean Absolute Deviation (MAD) 1/n Σ x i  x Example: Class =8 Class =26 9
10 Alternatively, we can take the square of the deviations from the mean to avoid the problem of the sum of the deviations equaling zero. Two measures of dispersion incorporate squares. Variance Σ (x i  x ) 2 To control for the number of scores involved, we can divide this sum by N, the number of observations. Population Variance: 1/n Σ (x i  x ) 2 = σ 2 Sample Variance: 1/(n1) Σ (x i  x ) 2 = s 2 Note that we divide the sample variance by (n1) rather than n. This has advantages when we go from a sample to population. Samples are less likely to contain extreme values and therefore may underestimate the amount of dispersion or variability in a set of measurements. When have information on entire population, replace (n1) with n, the actual population size. Example Class 1 s 2 = ( )/(61) = 16/5 = 3.2 Class 2 Properties of Variance 1. Approximates the average of the squared distances from the mean. 2. Units of measurement for the variance are the squares of those for the original data. 10
11 Standard Deviation The square root of the variance is called the standard deviation. Population Standard Deviation: σ = square root [1/(n) Σ (x i  x ) 2 ] Sample Standard Deviation: s = square root [1/(n1) Σ (x i  x ) 2 ] Example Class 1: 3.2 = = s Class 2: Interpretation: As a measure of variability, standard deviation tells us how much each score, in a set of scores, on the average, varies from the mean. The greater the variability around the mean of a distribution, the larger the standard deviation. Alternative way to calculate the variance and standard deviation using sum of squares 2 2 ( X X ) Variance = s = = X n 1 n 1 n ( X ) 2 2 ( X X ) SD = s = = X n 1 n 1 n ( X ) 11
12 Example: Class 1 Variance Obs X X Variance = s 2 = (1/5) * [310  (42) 2 / 6 ] = (1/5) * [ ] = 3.2 Class 2 Properties of Standard Deviation 1. s>=0. 2. Greater the variation, larger the value of s. 3. s is less than the maximum value of x i  x S < x i  x 4. Just as with mean, s can be affected by outliers, particularly for small data sets. 12
13 Obtaining this information from SPSS: Analyze Descriptive Statistics Frequencies Select Variable(s) Statistics N Mean Median Mode Std. Deviation Variance Range Minimum Maximum Percentiles Valid Missing Statistics CLASS1 CLASS a a. Multiple modes exist. The smallest value is shown Common graphical displays of dispersion within a data set. Box Plot SPSS: Graphs Boxplot Select summaries of separate variables and simple boxplot define select variable you want to plot Hit okay. Graphical summary of both central tendency and variation of a data set. Box contains central 50% of distribution, from lower quartile to upper quartile. Median marked by line within box. Lines from box are called whiskers. Extend to maximum and minimum values, unless there are outliers N = 6 CLASS1 6 CLASS2 13
14 Z SCORES Sometimes we want to make comparisons between observations from different data sets. For example, in our hypothetical classes, we may want to determine whether the student who received a score of 6 in class 1 did better or worse than the student who received a score of 9 in class 2. We are comparing two data sets with different means and standard deviations. To do this, we convert the means and standard deviations into standard scores also known as z scores. z = (X  x ) / s where X = particular observation x = mean of observations s = standard deviation of observations The z score describes how many standard deviations from the mean a score is located. Example: Class 1: X=6; x =7 ; s=1.79 Z=(67)/1.79 = Class 2: X=9; x =13 ; s=5.87 Z=(913)/5.87 = We conclude that the student in class 1 did slightly better than the student in class 2 based on these standard scores. You can transform all the scores in the classes into standard form or z scores. CLASS 1 ( x =7 ; s=1.79) CLASS 2 ( x =13 ; s=5.87) Obs X z Obs X z /1.79 = /5.87 = /1.79 = /5.87 = /1.79 = /5.87 = /1.79 = /5.87 = /1.79 = /5.87 = /1.79 = /5.87 = The scores for the two classes are now expressed in the same scale, and each class has a mean of 0 and a standard deviation of 1. 14
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