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Measures of Center and Dispersion

Measures of Center

A single value that summarizes a set of data. It locates the center of the values.

Arithmetic Mean - Population mean is a Parameter of the population. Sample mean is a Statistic and is frequently used to estimate the population mean. To calculate the mean total all of the data values and divide that total by the number of data points.
  • Properties of the arithmetic mean
  • Interval and ratio data sets have an arithmetic mean
  • All of the values are used in computing the mean
  • A set of data has only one mean. (Unique)
  • Can be used to compare two or more populations.
  • Only measure of central tendency where the sum of the deviations of each value from the mean will always be zero.
Median - The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest. There are as many values above the median as below it in the data array.
  • Properties of the median
  • The median is unique. Only one for a given set of data.
  • To determine order the data from low to high or high to low and find the middle value
  • It is not affected by extremely large or small values.
  • It can be computed for an open-ended frequency distribution, if the median does not lie in the open-ended class. (An open ended class is one that does not have a specific limit. An example would be a class that is stated as: $100,000 or more. How much is "or more"? No one knows where the class ends. This makes it open ended)
  • It can be computed for ratio, interval and ordinal level data.
Mode - The value of the observation that appears most frequently.
  • Properties of the mode
  • Can determine the mode for all levels of data.
  • Not affected by extremely high or low values.
  • Can be used with open-ended distributions.
  • A data set may not posses a unique mode. No mode, one, two, three or more are all possibilities.

Measures of Dispersion


Calculations - Data NOT IN a Frequency Table

The formulas shown above tell you how to calculate the mean, variance and standard deviation if you have data that is NOT in a frequency table. To illustrate the use of the formulas assume that we sample the amount of time people wait to get a web connection using their dial-up phone line. If we collect a sample of size 5 (n=5) the data (values of x) in seconds might look as follows: 10, 24, 16, 20 and 10.

To calculate the sample mean we would add all of the values 10+24+16+20+10 and then divide the result by 5 (the number of data points). Since our sum is 80 and we divide it by 5 the mean is 16 seconds.

To calculate the variance or standard deviation when the data is NOT IN A FREQUENCY TABLE we have a choice of two formulas (only one shown for the variance). We will only illustrate the use of formula 1. Using Formula 1 we need to:

  1. Square each of the x values
  2. Find the total of those squares
  3. Multiply the previous total by the number of data points
  4. Go to the x column and total it
  5. Square the total of the x column
  6. Subtract the square of the x column from the sum of the x's squared (from step 3)
  7. Divide the result from step 5 by the product of the sample size times the sample size minus 1 [n*(n-1)]
  8. Take the square root of the above result

As an example we would

  1. Square each of the x's which gives: 100,576, 256, 400 and 100.
  2. Add all of the x squares which gives: 1,432
  3. Multiply 1,432 by 5 (the number of data points) giving: 7160
  4. Add of of the x's which gives:80
  5. Square the 80 from the previous step: 6,400
  6. Subtract step 5's result from step 3's result: 7160 - 6400
  7. Divide 6400 by 5(5-1) giving:320 (Variance)
  8. Take the square root or 320 giving:17.89 (Standard deviation)

Practice Question 1 - Data not in a freqency table - Assume that you take a sample of the length of time that people wait in line at an ATM machine and record the following values (in minutes): 3, 5, 2, 7, 3. Use this data to calculate the sample mean, standard deviation and variance. (answers at the end of this section)


Calculations - Data IN a Frequency Table

The formulas shown above tell you how to calculate the mean, variance and standard deviation if you have data that is in a frequency table. To illustrate the use of the formulas assume that we take a new and larger sample of the amount of time people wait to get a web connection using their dial-up phone line. Suppose we collect a sample of size 50 and get the data shown in the first and second columns on the table shown below.

Time Waiting for Service
Class (Time) Frequency Mid-Point Freq*Mid-Point Freq*Mid-Point2
10 - 1420122402880
15 - 1910171702890
20 - 2420224409680
Totals50No Meaning722500772500

In the table the first column represents our classes (time), the second column is the frequency for each class, the third column is the class midpoint (upper class limit plus lower class limit divided by 2), the next column is the frequency times the class midpoint and the last column is the class frequency times the square of the class midpoint (ONLY THE CLASS MIDPOINT IS SQUARED!).

To calculate the mean we total the column where we multiplied the class mean times the class frequency: 240 + 170 + 440 = 850. Now you divide 850 by the total the frequency column - 50. The result is 850/50 = 17. This is the mean of our sample data.

To calculate the standard deviation:

Practice Question 2 - Data in a freqency table - Assume that you take a sample of the customers at a local pizza parlor. The data you collect is shown below. Use the data to calculate the sample mean, standard deviation and variance. (answers at the end of this section)
Ages of Customers
Customer age Frequency
0 - 1923
20 - 2917
30 - 3918
40 - 4915
50 - 6912

Interpretation and Uses of the Standard Deviation

Chebyshev's Theorem - For any set of observations (sample or population), the minimum proportion of the values that lie within k standard deviations of the mean is at least 1 - 1/k2, where k is any constant greater than 1. For example: assume that the mean of a distribution is 20 and the standard deviation is 5. You want to know the fraction of the data that can be found between 10 and 30. Using a z score format of z = (x - mean)/std dev. you have: (20-10)/5=2 and (30-20)/5=2 so your data points are 2 standard deviations above and below the mean. Using Chebyshev's rule you have % = 1 - 1/22. Simplifying you have % = 1 - 1/4 = 3/4 = 0.75 or 75%. 75% of the data will lie between 10 and 30.

The Empirical Rule

- For a symmetrical, bell-shaped frequency distribution, approximately 68 percent of the observations will lie within plus and minus one standard deviation of the mean; about 95 percent of the observations will lie within plus and minus two standard deviations of the mean; and practically all ( 99.7 percent ) will lie within plus and minus three standard deviations of the mean.

Problems using the Empirical Rule (bell-shaped symmetric distribution)

1.0 Given: A normal distribution with a mean of $50 and a standard deviation of $5

The values of $40 and $60 are plus and minus two standard deviations from the mean. Looking at the Empirical Rule figure (above) you find that 95% of the population will be within plus or minus two standard deviations of the mean.

2.0 Given: A normally distributed population has a mean of 35 and a standard deviation of 3.

Measures of Position

- Z Score, Quartiles, Deciles, and Percentiles

All of these measures address the issue of where the data point is in relation to the rest of the data. Z Score tells you how many standard deviations above or below the mean a given value of x is located. Quartiles divide the data into quarters, deciles divide the data into tenths and percentiles divide the data into hundredths.