DIS - Measures of Dispersion (Spread) Lesson

Measures of Dispersion (Spread)

Measures of Dispersion deal with the spread ada-reference of the data. The spread of a distribution refers to the variability of the data. If the data cluster around a single central value, the spread is smaller. The further the observations fall from the center, the greater the spread or variability of the set. (range, interquartile range, Mean Absolute Deviation, and Standard Deviation measure the spread of data). It also deals with the range.

The range of a set of data is the difference between the greatest data element and the least data element. The last dispersion measurement is quartiles.

There are three quartiles ada-reference of a data set which divide the set into four equal parts, when listed in numerical order.

A measure of dispersion called the Interquartile Range (IQR) is the difference between the third quartile ( equation image indicator LaTeX: Q_3 $Q_3$ ) and the first quartile ( LaTeX: Q_1 $Q_1$ ). This IQR measures the spread of the middle half of the data. Remember, the five-number summary (smallest value, first quartile, median, third quartile, largest value) is used to make box and whisker plots.

The variance of data is the average of the squares of the deviations of the observations from their mean (each data point minus the mean, then squared) and is given by $LaTeX: \frac{\left(x_1-\overline x\right)^2+\left(x_2-\overline x\right)+...+\left(x_n-\overline x\right)^2}{n}$ $\frac{\left(x_1-\overline x\right)^2+\left(x_2-\overline x\right)+...+\left(x_n-\overline x\right)^2}{n}$ where n is the number of data points, equation image indicator $LaTeX: \overline x$ $\overline x$ is the sample mean, and LaTeX: x_1,x_2,x_3,,......,x_n $x_1,x_2,x_3,,......,x_n$ are the data points.

The other measure of dispersion is the square root of the variance, called the Standard Deviation. The greater the standard deviation the more spread out the data are. The symbol for Standard Deviation is the Greek letter σ ('sigma') and is given by the formula equation image indicator $LaTeX: \sigma=\sqrt[]{\frac{\left(x_1-\overline x\right)^2+\left(x_2-\overline x\right)^2+...\left(x_n-\overline x\right)^2}{n}}$ $\sigma=\sqrt[]{\frac{\left(x_1-\overline x\right)^2+\left(x_2-\overline x\right)^2+...\left(x_n-\overline x\right)^2}{n}}$ . Because the numerator is a sum of numbers, you may see the formula written equation image indicator $LaTeX: \sigma=\sqrt[]{\frac{1}{n}\sum\left(x_i-\overline x\right)^2}$ $\sigma=\sqrt[]{\frac{1}{n}\sum\left(x_i-\overline x\right)^2}$ where ∑ (capitol 'sigma') means the sum from 1 to n.

Given the following data set of values: 10, 2, 38, 23, 38, 23, 21, 48, 92

equation image indicator $LaTeX: Mean\:\:x=\frac{10+2+38+23+38+23+21+48+92}{9} \\ x\approx 32.78 \\ Median: 2, 10, 21, 23, \color{blue}23\color{black}, 38, 38, 48, 92 \\ Mode: 23, 38 \\ Range: 92-2=90 \\ IQR \text{: }\:Q_1=\frac{10+21}{2}=15.5 \\ Q_3=\frac{38+48}{2}=43 \\ Q_3-Q_1=43-16.5=27.5 \\ \text{Standard Deviaion } (\sigma ) \text{ Computed from a calculator 24.92}$ $Mean\:\:x=\frac{10+2+38+23+38+23+21+48+92}{9} \\ x\approx 32.78 \\ Median: 2, 10, 21, 23, \color{blue}23\color{black}, 38, 38, 48, 92 \\ Mode: 23, 38 \\ Range: 92-2=90 \\ IQR \text{: }\:Q_1=\frac{10+21}{2}=15.5 \\ Q_3=\frac{38+48}{2}=43 \\ Q_3-Q_1=43-16.5=27.5 \\ \text{Standard Deviaion } (\sigma ) \text{ Computed from a calculator 24.92}$

Computing the Interquartile Range of Data Sets (Video 1 and 2)

Video 1 explores an example that computes the Interquartile Range of a data set with an odd number of data points.

Video 2 explores an example that computes the Interquartile Range of a data set with an even number of data points.

Computing the Interquartile Range of Data Sets (Video 12, 3, and 4)

We discussed in Lesson 1 the Measures of Central Tendency. Now we will build upon those measures to include the Measures of Dispersion or Spread, which show us how far or close the data points are from the center. Videos 1 and 2 introduce measures of dispersion, and videos 3 and 4 apply these measures and compare sets of data using these measures.

Watch the following videos that explore examples of measures of dispersion (spread).

In the first 3:09 of video 1 an example (numbers 1 and 2) is shown which compares two data sets and then draws conclusions from the means and ranges. From 3:10 to the end of the video, an example with 6 parts (numbered 3-8) computes the mean, median, range, lower quartile, upper quartile, and inner quartile range (IQR) of two data sets. These are computed by hand and with a TI-84 Plus Silver Edition calculator.

In the first 10:14 of video 2 an example (numbers 1 and 2) is shown which compares two sets of test scores and then computes the mean, median, range, lower quartile, upper quartile, and inner quartile range (IQR). It then draws conclusions from comparing the two data sets. From 10:15 to the end of the video, an example with 3 parts (numbered 6a, 6b, and 6c) computes the Mean Deviation, Variance, and the Standard Deviation of the data. These are computed by hand and with a TI-84 Plus Silver Edition calculator.

In the first 2:00 of video 3, two examples (numbered 7 and 8) are shown what the difference in the measures of variability (spread) suggest, and draws conclusions from the results. From 2:01 to 6:08, four minor examples are shown that computes the standard deviation of sets of data. From 6:09 to 7:40, an exploration is shown creating a data set that has a given standard deviation. From 7:41 to the end of the video, two examples are shown that compute the standard deviation and range of data sets from a table of values.

In the first 1:30 of video 4, an example is shown that computes the standard deviations from four dot plots. From 1:31 to 7:23, the mean, median, and mode of a set of data are computed. Next, the range, absolute mean deviation, and the standard deviation are computed, followed by a dot plot of the data being created. From 7:24 to the end of the video, one more example is explored that computes the mean, median, mode, range, absolute mean deviation, and the standard deviation of a data set. Two dot plots are created, and then the centers and spread are discussed.

Population vs. Sample Statistics Introduction

In the following video, the population mean, sample mean, population standard deviation, and the sample standard deviation are introduced.

Measures of Spread - Mean, Variance, Standard Deviation, and Range

In the first 4:34 of the following video, the mean and range are explored. From 4:35 to 10:00, the variance is explored. From 10:01 to the end of the video, the standard deviation is explored.

Variance (Videos 1 and 2)

Watch the following videos exploring variances.

In video 1, the sample variance is explored.

In video 2, the population variance is explored.

Population Statistics

In the following video, the population mean, population variance, and population standard deviation are explored.

Application: Measures of Central Tendency and Dispersion

In the following video, an example is shown that computes the mean, median, standard deviation, interquartile range of a set of data.

Measures of Spread Practice

Now it is time to explore some more examples of measures of spread.

Measures of Data Practice

Now it is time for you to practice some measures of spread problems.

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