Res-Econ TBL 4 Notes - Measures of Central Tendency - Mean, Median, Mode, Outlier Measures of Dispersion PDF

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Measures of Central Tendency - Mean, Median, Mode, Outlier
Measures of Dispersion -Variance, Standard Dev., Range, Mean Absolute Deviation, Coefficient of Variation
Percentiles and Quartiles
Definitions, Steps, and Formulas provided.
Professor: Wayne Roy Gayle
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Resource Economics 212 September 28 2016 Lecture/Textbook Notes TBL 4 Measures of Central Tendency  Central Tendency o The middle or typical values of a distribution o Data on a graph can be skewed left, skewed right, or symmetric.  The Mean o The sum of the data values divided by the number of data items.  For population: Called the mean or expected value or average  For sample: Called the sample mean or sample average  Excel Command: =AVERAGE(Data) o = (x1+x2+x3+x4…)/total number of data items  The Median o The midpoint of a set of sorted data.  Separates the upper and lower half of the sorted observations  Denotes the 50th percentile o Calculating the median:  If the number of observations, n, is odd, then the median is the middle observation in the sorted data.  If the number of observations, n, is even, then the median is the average of the middle two observations in the sorted data.  Excel Command: =MEDIAN(DATA)  Position of Median:  Center value that divides ordered data into two halves  (Number of observations + 1) / 2  The Mode o The most frequently occurring data value.  Value that is most frequent in the data  The frequency and relative frequency are the highest among all values  A data set may have multiple modes or no mode at all.  Excel Command: =MODE(Data)  Outlier o A value that is higher or lower than the rest of the data values in an extreme way o Effects:  Mean: affected, since it will be calculated into the mean  Median: not affected, since the values do not matter for the median  Mode: not affected, since an outlier wouldn’t be the most frequent one. Measures of Dispersion  Dispersion

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o Measures the level of variance in the data. o The spread of the data points about the center of the distribution of the data. Variance o Average of the squared distances between the data values and their mean. o Conceptualize:  Measure difference between each observation to the mean  Square each of the differences  Sum the squared differences and divide by the appropriate factor  Excel Function: =VAR(Data) o Calculations  Population Variance  σ2 = Σ (Xi - μ )2 / N  Sample Variance  s2 = Σ (xi - x )2 / ( n - 1 ) Standard Deviation o The square root of the variance o Why use it?  For unit matching  Excel Function: =STDEV(Data) Range o Difference between the maximum and minimum values in a data set. o Range = X(max) – X(min)  Very sensitive to outliers  Excel Command: =MAX(Data) – MIN(Data) Mean Absolute Deviation (MAD) o Measures the average of the absolute from the center.  Absolute values must be used otherwise the deviation around the mean would sum to zero  Less sensitive to outliers relative to the Range, Variance. o Calculations N

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MAD=

∑ |x i−´x| i=1

n  Excel Function: =AVEDEV(Data) Coefficient of Variation o Useful for comparing variables measured in different units or with different means. o A unit-free measure of dispersion. o Expressed as a percent of the mean. o Only appropriate for nonnegative data. It is undefined if the mean is zero or negative. o Calculation:  CV = 100 * s/ ´x

Percentiles  Percentiles o A value below which a certain percentage of the data fall.  55th percentile: the value below which 55% of the data falls. o Percentiles divide data into equal chunks  Quartiles: 3 values that divide the data into 4 equal chunks  Deciles: 9 values that divide the data into 10 equal chunks o Computing Quartiles  Sort the raw data in ascending order (From low to high)  Q1 is at position 0.25(n+1)  Q2 is at position 0.50(n+1)  Q3 is at position 0.75(n+1)  Q1 leaves 25% of the values below it (it is the 25th percentile)  Q2 is the median (50th percentile)  Q3 leaves 75% of the values below it (75th percentile)  Five Number Summary: o 1. Minimum Value o 2. Q1 o 3. Q2 o 4. Q3 o 5. Maximum Value  Interquartile Range o The range within which the middle 50% of the data lie.  IQR = Q3 – Q1  Quartiles – Key Distribution Shapes o Uniform o Bell-Shaped o Right Skewed o Left Skewed

TBL 4 Class  Work on project with group and divide the tasks  Made excel template to calculate mean, median, standard deviation, and quartiles.  IRAT was on TBL 4....