In 3-8, find the mean, the median, and the mode for each set...

In $3-8,$ find the mean, the median, and the mode for each set of data.
$$
\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 5 & {6} \\ {4} & {10} \\ {3} & {15} \\ {2} & {11} \\ {1} & {2} \\ {0} & {1} \\ \hline\end{array}
$$

Key Concepts

Frequency Distribution

A frequency distribution is a summary of data that shows the number of times each value (or range of values) occurs in a dataset. It organizes raw data into a table or chart with values (or classes) and their corresponding frequencies, making it easier to analyze patterns and calculate statistical measures such as mean, median, and mode.

Mean

The mean is the arithmetic average of a dataset, calculated by summing all the values multiplied by their frequencies and then dividing by the total number of observations. In the context of frequency distributions, it is effectively a weighted average where each value contributes proportionally to its frequency.

Median

The median is the middle value of an ordered dataset. For a frequency distribution, it is determined by locating the point in the cumulative frequency where half of the observations have been accounted for. The median effectively splits the data into two equal halves.

Mode

The mode is the value that appears most frequently in the dataset. In a frequency table, it is the data point associated with the highest frequency, indicating the most common occurrence in the dataset.

Transcript

00:01 And a list of various x's and how many times they've occurred frequencies.

00:06 And so for number three, we're asked to find the mean, median, and mode.

00:10 In order to do that, we need to, first of all, add up all the frequencies.

00:15 So we're going to add up six plus 10, plus 15, plus 11, plus 2, and plus 1.

00:23 And we're going to get 45.

00:26 So go ahead and write 45 down.

00:30 Then we need to create another column where we're going to be multiplying the numbers in your list by the number of frequencies it occurred.

00:40 So five times six, for example, is going to be our first one.

00:44 So you're going to get a 30.

00:47 Four times 10 is 40.

00:50 Three times 15 is 45.

00:56 Then you've got two times one is...

00:58 Oh, it's 11.

00:59 Sorry.

01:00 That's an 11 there.

01:01 2 times 11 is 22.

01:09 Then we got 2 and we've got 0.

01:12 So we're going to add 30 plus 40 plus 45 plus 22 plus 2 and that's going to get us 139.

01:23 So in order to get the mean, we are going to need to divide the big number divided by the little number.

01:31 So 139 over 45.

01:38 And so when you do 139 divided by 45, you're going to get 3 .0 and then 8 is going to be repeating.

01:50 Now, if your teacher wants you to round the nearest 10th, that would be 3 .1, or if you're running into two decimal places, that will be 3 .09.

01:58 So the next thing we are asked to do is to find the median...

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