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Measures of Central Tendency

Measures of Central Tendency. MARE 250 Dr. Jason Turner. Centracidal Tendencies. Three most important: Mean Median Mode quantitative & qualitative. quantitative. Mean Girls. Mean –. Mean – the average X i = measurement in a population μ = mean of a population

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Measures of Central Tendency

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  1. Measures of Central Tendency MARE 250 Dr. Jason Turner

  2. Centracidal Tendencies Three most important: Mean Median Mode quantitative & qualitative quantitative

  3. Mean Girls Mean – Mean – the average Xi = measurement in a population μ = mean of a population N = size of the population Σ = summation; in this case it indicates that all X values are to be added together, and the sum divided by the population size N Calculation of a Population mean: (Zar eq. 3.1)

  4. Mean Streets Mean – the average Xi = measurement in a population x = mean of a sample n = size of the sample ∑ = summation Calculation of a Sample mean: (Zar eq. 3.2)

  5. 927 mm = 92.7 mm 10 For Example Calculating a Sample Mean A sample population of juvenile mahi-mahi Xi (in mm): 52, 65, 78, 79, 85, 102, 110, 115, 116, 125 ∑ Xi = 927 mm n = 10

  6. The Median First – arrange measurements in order of magnitude – ascending or descending M = median (Zar eq. 3.4) if the sample size n is an odd number.

  7. The Median – How too... If the number of observations is ODD, then the median is the observation exactly in the middle of the ordered list If the number of observations is EVEN then the median is the mean of the two middle observations in the ordered list In both cases, if we let n denote the number of observations, then the median is at position (n + 1)/2 in the ordered list (Zar eq. 3.4) if the sample size n is an odd number.

  8. For Example Calculating a Median with Odd sample size A sample population of juvenile mahi-mahi Xi (in mm): 65, 78, 79, 85, 102, 110, 115, 116, 125 n = 9 M = X (n+1)/2 = X (9+1)/2 = X 5 = 102 However...

  9. Evenflow Calculating a Median w/ even sample size If the sample size n is an even number, then there is no ‘middle’ value, and M is the average of the two middle values Xi (in mm): 52, 65, 78, 79, 85, 102, 110, 115, 116, 125 n = 10 M = [X (10/2) +X (10/2) +1]/2 M = [5 + 6]/2 M = X 5.5 = 93.5 And Yet...

  10. In the event of a tie… Calculating a Median w/ even sample size In this case where the median value is within a tied set of values (a set of equivalent observations), the median is often interpolated to generate a more accurate estimate. To interpolate means to insert something into a set of existing things; interpolation in math refers to estimate a value between existing values in a case where you need a more accurate value than those given.

  11. In the event of a tie… Calculating a Median w/ even sample size In this case, to generate a more accurate estimate, the median is calculated as: Zar eq. 3.5 Cum. freq. in this case refers to the cumulative frequency (in numbers, not percent) of the previous classes.

  12. In the event of a tie… Calculating a Median w/ even sample size Xi (in mm): 52, 65, 78, 79, 85, 90, 90, 90, 90, 102, 110, 115, 116, 125 n = 14 M = (89.5) + [0.5(14)-5/4] (1) M= 90

  13. 30 20 Number of Individuals 10 0 The Mode The mode is typically defined as the most frequently occurring measurement in a set of data The mode is useful if the distribution is skewed or bimodal (having two very pronounced values around which data are concentrated)

  14. Reach out and Touch Faith First – obtain the frequency of occurrence of each value and note the greatest frequency If the greatest frequency is 1 (i.e., no value occurs more than once), then the data set has no mode If the greatest frequency is 2 or greater, then any value that occurs with that greatest frequency is called a mode of the data set

  15. The Mode In the previous example… Xi (in mm): 52, 65, 78, 79, 85, 90, 90, 90, 90, 102, 110, 115, 116, 125 n = 14 The mode is 90

  16. You are so totally skewed! The mean is sensitive to extreme (very large or small) observations and the median is not Therefore – you can determine how skewed your data is by looking at the relationship between median and mean Mean is Greater than the Median Mean and Median are Equal Mean is Less Than the Median

  17. Resistance Measures A resistance measure is not sensitive to the influences of a few extreme observations Median – resistant measure of center Mean – not Resistance of Mean can be improved by using – Trimmed Means – a specified percentage of the smallest and largest observations are removed before computing the mean Will do something like this later when exploring the data and evaluating outliers…(their effects upon the mean)

  18. How To on Computer On Minitab: Your data must be in a single column Go to the 'Stat' menu, and select 'Basic stats', then 'Display descriptive stats'. Select your data column in the 'variables' box. The output will generally go to the session window, or if you select 'graphical summary' in the 'graphs' options, it will be given in a separate window. This will give you a number of basic descriptive stats, though not the mode.

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