Looking For Patterns in Grouped Data

Script Ohio by flickr user Junior Sam

Looking For Patterns in Grouped Data

Script Ohio by flickr user Junior Sam

Standard Deviation (σ): What's the difference between "σ" and "s"?

The symbol for standard deviation of a population or large sample is "σ" (sometimes written as "σ "), and the symbol for standard deviation of a sample is 's'. A large sample is defined as a sample with 30 or more data items. In this course, we will use only "σ" (sigma), which represents the standard deviation of the population.

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Let's take a look at a visual explanation of why we calculate the standard deviation this way ...

Measures of Dispersion (Variability)determine how "spread out" or varied" a set of data is.

http://www.seeingstatistics.com/

Working with Grouped DataA frequency distribution table shows the number of elements of data (frequency) at each measure. Sometimes the measures need to be grouped, especially if the measures are continuous.

Determine the mean, median, mode, range, and standard deviation for the student heights.

height interval interval mean # of students153.5 to 160.5 157 5160.5 to 167.5 164 16167.5 to 174.5 171 43174.5 to 181.5 178 27181.5 to 188.5 185 9 Total 100

Example: The table below is a frequency distribution table that shows the heights of 100 Senior 4 students. The students are grouped into suitable height groups in 7 cm. intervals.

mean = 172.33 median = 171.00 mode = 171.00

range = 28 standard deviation = 6.837

Determine the mean, median, mode, range, and standard deviation for the student heights.

height interval interval mean # of students153.5 to 160.5 157 5160.5 to 167.5 164 16167.5 to 174.5 171 43174.5 to 181.5 178 27181.5 to 188.5 185 9 Total 100

mean = 172.33 median = 171.00 mode = 171.00

range = 28 standard deviation = 6.837

Working with Grouped DataA probability distribution table shows the percent of elements of data (probability) of each measure. Sometimes the measures need to be grouped, especially if the measures are continuous.

height interval interval mean % of students153.5 to 160.5 157 0.05160.5 to 167.5 164 0.16167.5 to 174.5 171 0.43174.5 to 181.5 178 0.27181.5 to 188.5 185 0.09 Total 1

Example: The table below is a frequency distribution table that shows the heights of 100 Senior 4 students. The students are grouped into suitable height groups in 7 cm. intervals.

Grouped Data and Histograms A histogram is a bar graph that shows equal intervals of a measured or counted quantity on the horizontal axis, and the frequencies associated with these intervals on the vertical axis.

Drawing a histogram is useful because it shows the distribution of the heights of the students.

A histogram is known as a Frequency Distribution Graph when the data is obtained from a frequency distribution.

A histogram is known as a Probability Distribution Graph when the data is obtained from a probability distribution.

Learn more about constructing a Histogram.

Click: Contents > 2. Seeing Data > 2.3 Histogramhttp://www.seeingstatistics.com/

Let's apply what we've learned ...

The frequency distribution table at right shows the midterm marks of 85 Grade 12 math students. The first column shows the mark interval, the second column the average mark within each mark interval, and the third column the number of students at each mark.

(a) Calculate the mean and std. dev. to two decimal places.

(b) Calculate the number of students that have marks within one std. dev. of the mean.

(c) What percent of students have marks within one std. dev. of the mean?

58 or 70

68% or 82%

Mean = 66.99 σ = 13.27

mark interval mark # of students 29 to 37 33 1 38 to 46 42 4 47 to 55 51 12 56 to 64 60 18 65 to 73 69 24 74 to 82 78 16 83 to 91 87 7 92 to 100 96 3

Total 85

This is the correct way to do this.

An experiment was performed to determine the approximate mass of a penny. Three hundred pennies were weighed, and the weights were recorded in the frequency distribution table shown below.

Mass (grams) 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4Frequency 2 4 34 71 94 74 17 4

Determine the mean, median, mode, range, and standard deviation of the data. Create a histogram that shows the frequencies of different masses of this set of pennies.

HOMEWORK

Four hundred people were surveyed to find how many videos they had rented during the last month. Determine the mean and median of the frequency distribution shown below, and draw a probability distribution histogram. Also, determine the mode by inspecting the frequency distribution and the histogram.

HOMEWORK

1 28 2 102 3 160 4 70 5 25 6 13 7 0 8 2

No. of Videos Returned No. of Persons

The table shows the weights (in pounds) of 125 newborn infants. The first column shows the weight interval, the second column the average weight within each weight interval, and the third column the number of newborn infants at each weight.(a) Calculate the mean weight and standard deviation.

(d) What percent of the infants have weights that are within one standard deviation of the mean weight?

(c) Determine the number of infants whose weights are within one standard deviation of the mean weight.

(b) Calculate the weight of an infant at one standard deviation below the mean weight, and one standard deviation above the mean.

HOMEWORK weight interval mean interval # of infants 3.5 to 4.5 4 4 4.5 to 5.5 5 11 5.5 to 6.5 6 19 6.5 to 7.5 7 33 7.5 to 8.5 8 29 8.5 to 9.5 9 17 9.5 to 10.5 10 8 10.5 to 11.5 11 4

TOTAL 125