Day 4: Statistics Review

Note: I have a personal webpage that I use for class -- that is, I use it when I know that students are using it. I spent a good part of the evening updating the page with information about a test on Statistics that we're having on Friday. I copied over the code to the blog. No reason that it can't count toward my daily updates.

STATISTICS REVIEW

We are having an important Statistics exam on Friday, May 2, 2014.
Statistics will be an important part of the Common Core Algebra Exam.
You should review this material.

Measures of Central Tendency: The mean, median and mode. They are numbers used to describe a set of data. See the individual definitions below.

Mean: The average. To find the mean, add up all the data and divide by the number of data items there are. Example: The mean of 6, 3, 5, and 12 is (6+3+5+12)/4 = 26/4 = 6.5.

Median: The middle number when the data are in order. If the data are not in order, than the number in the ”middle” is meaningless. If there are two numbers in the middle, the median is the average of the two numbers. Example: The median of 6, 3, 5, and 12 would be found be first ordering the numbers as 3, 5, 6, 12. The numbers 5 & 6 are in the middle, so the median is 5.5.

Mode: The most frequently appearing piece of data. Unlike mean or median, there can be more than one mode. There might be no mode if no data is repeated. Mode doesn’t have to be a number: consider the data set (red, red, yellow, blue, blue, blue); “blue” is the mode. Examples: (1, 2, 3, 3, 3, 4, 4): “3” is the mode; (88, 95, 92, 95, 88): 88 and 95 are the modes; (7, 8, 9, 10): no mode.

Frequency The number of times something occurs. In the data (10, 10, 11, 11, 11, 12, 12, 13), “11” has a frequency of 3, and the total frequency of the set is 8.

FREQUENCY TABLES

A Frequency Table is a summary of the data, organized as a table. The data are on in the left column. The frequency of each piece of data is one the right.
There are three examples of frequency tables below.

In the first example, every number in the data is listed, along with its frequency. The interval of the table is 1. Because of this we can find the mean, median and the mode of the data. We can re-create the data if we wanted to by writing out all the numbers.

In the second example, the data is collected into intervals of 10. We don’t know the actual numbers; we only have approximate information. Because of this we cannot find the exact mean, median or mode. However, we can find which interval contains the median and which interval is the most frequent. (There are methods to find a mean, but we aren’t going to calculate that right now.)

In the third example, the data aren’t numbers, they’re adjectives (qualitative data). Because of this, we can find a mode, but not a mean or median.

Finding the Mode, Median and Mode of a Frequency Table

In the first table, the total frequency (the sum of the Frequency column) is 20. If we write out the data, we can see that it has a total of 36. Divide 36 by 20 and we get a mean of 1.8. We can also see that there are more 2s than any other number, so 2 is the mode. And out of 20 numbers, the 10th and 11th are in the middle: both of those numbers are 2, so the median is 2.

However, we don’t need to rewrite all the data. (And if the table were bigger, we wouldn’t want to write out all the data!) The number with the highest Frequency is 2, with a frequency of 7 (mode). If you keep a running count as you go down the Frequency column, you will see that the 9th through 15th numbers are all 2 – that includes the 10th and 11th number (median). And if we know how many of each number there are, we can take a short-cut to get the sum of the data: (0 x 3) + (1 x 5) + (2 x 7) + (3 x 4) + (4 x 0) + (5 x 1) = 36. Divide 36 by 20 and we get 1.8 (median).

In the second table, the mode and median can be found using the same method.

In the third table, because the data are descriptive and not numeric (quantitative), there is no median. A middle size somewhere between Medium and Large wouldn’t have any meaning.