3.2 Measures of Dispersion
What types of values are the units of measure in variance terms?
Squared values. For instance, if the variable is measured in dollars, the variance is measures in dollars squared. This makes interpreting the variance difficult.
What is the second measure of dispersion?
Standard Deviation
What can be used to determine the percentage of data that will lie within, k, standard deviations of the mean with a bell-shaped distribution?
The Empirical Rule
Range
The range, R, of a variable is the difference between the largest and smallest data value. That is, Range = R = Largest Data Value - Smallest Data Value
How is the sample variance calculated?
By squaring the sample standard deviation.
The population standard deviation is symbolically represented by...
o' (lowercase Greek sigma)
What does standard deviation represent?
the "typical" deviation from the mean... In other words, the SD may be used to judge whether a particular observation is "far away" from the mean of a data set.
Steps to calculating variance for a population through StatCrunch
1. Open StatCrunch 2. Enter your data 3. Click on Stats > Summary Stats > Columns 4. Select Population Column 5. Under Statistics, select "Variance" 6. Compute!
Steps to calculate Population SD through StatCrunch
1. Open StatCrunch. 2. Enter your data. 3. Click on STAT > Summary Stats > Columns 4. Select the column containing the data. 5. Under statistics, select "Unadj. std. dev." 6. Compute!
Steps to calculate Sample SD through StatCrunch
1. Open StatCrunch. 2. Enter your sample data. 3. Click on STAT > Summary Stats > Columns 4. Select the column containing the sample data. 5. Under statistics, select "std. dev." 6. Compute!
Is a measure of 27 inches "far away" from a mean of 18 inches? Suppose the data come from a sample whose standard deviation is 3 inches. How many standard deviations is 27 inches from 18 inches?
27 inches is THREE standard deviations away from 18 inches
What can be said about a set of data with a standard deviation of 0?
All the observations are the same value. If all observations have the same value, then that value will also be the mean of the data. Therefore, the sum of the squared differences from the mean will be 0, and the standard deviation will be 0.
How is the population variance calculated?
By squaring the population standard deviation.
Identify the given statement as either true or false... The standard deviation is a resistant measure of spread.
FALSE! Since extreme values will increase the standard deviation greatly, the standard deviation cannot be a resistant measure of spread.
Identify the given statement as either true or false... The standard deviation can be negative.
FALSE! There is no way that the calculation of the population or sample standard deviation can produce a negative number. This makes intuitive sense because the standard deviation measures the spread of the data from the mean.
The Empirical Rule
If a distribution is roughly bell-shaped, then... Approximately 68% of the data will lie within 1 standard deviation of the mean. That is, approximately 68% of the data will lie between mu-1sigma and mu+1sigma Approximately 95% of the data will lie within 2 standard deviations of the mean. That is, approximately 95% of the data will lie between mu-2sigma and mu+2sigma Approximately 99.7% of the data will lie within 3 standard deviation of the mean. That is, approximately 99.7% of the data will lie between mu-3sigma and mu+3sigma
Suppose the SD of the underlying data is 7 in. Is 27 in. far away from a mean of 18 in.?
NO, because 27 in. is less than 2 SD away from 18 inches.
Is standard deviation resistant?
No.
To compute the range, the data must be...
Quantitative
What is the simplest measure of dispersion?
Range
Sample Standard Deviation
The sample standard deviation, s, of a variable is the square root of the sum of squared deviations about the sample mean divided by n-1, where n is the sample size. The form
What is the SD used to describe?
The spread in symmetric distributions (while the mean is used to describe the center of the distribution.
Population Standard Deviation
The square root of the sum of squared deviations about the population mean divided by the number of observations in the population, N. In other words, it is the square root of the mean of the squared deviations about the population mean.
Variance
The variance of a variable is the square of the standard deviation. The population variance is sigma squared, and the sample variance is s^2
What is important to remember when notating units in regards to variance?
To be sure to include the units squared when reporting the variance.
True or False: When comparing two populations, the larger the standard deviation, the more dispersion the distribution has, provided that the variable of interest from the two populations has the same unit of measure.
True, because the standard deviation describes how far, on average, each observation is from the typical value. A larger standard deviation means that observations are more distant from the typical value, and therefore, more dispersed.
What can one conclude about the dispersion of any particular set of data, based on a histograms spread?
Usually there is more dispersion with more spread out, and in symmetric graphs in comparison so a skewed graph.
What can one conclude about the dispersion of any particular set of data, based on standard deviation?
Usually, the greater the standard deviation, the more dispersion there is. This goes for Range, as well.
Over the past 10 years, five mutual funds all had the same mean rate of return. The standard deviations for each of the five mutual funds are shown below. Capitol Investment: 8.7% Vanity: 9.4% Global Advisor: 7.6% International Equities: 6.2% Nomad: 5.6% Which Mutual fund is least consistent in return?
Vanity
What is the third measure of dispersion?
Variance
Is 27 inches far away from a mean of 18 inches with a standard deviation of 3?
YES, because 27 in. is MORE THAN 2 standard deviations away from 18 inches
The sum of the deviations about the mean always equals...
Zero
