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.
$$
\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 12 & {7} \\ {11} & {15} \\ {10} & {13} \\ {9} & {16} \\ {8} & {16} \\ {8} & {14} \\ {7} & {15} \\ {6} & {9} \\ {5} & {2} \\ \hline\end{array}
$$

Key Concepts

Mean

The mean is a measure of central tendency calculated by summing all the observed values (or products of values and their frequencies in a weighted dataset) and then dividing this total by the number of observations or the sum of the frequencies. It provides an overall average of the data, which is sensitive to extremely high or low values.

Median

The median is the middle value in an ordered dataset, and it divides the data into two equal halves. When data is grouped with frequencies, it involves finding the cumulative frequency until the middle position is identified. The median is robust, meaning it is less affected by outliers compared to the mean.

Mode

The mode is the value that appears most frequently in a dataset. It represents the most common observation and can be used as an indicator of the distribution’s peak. In frequency distributions, the mode is identified by finding the value with the highest frequency.

Frequency Distribution

A frequency distribution is a tabular representation of data that shows how many times each value (or range of values) occurs. It organizes raw data into a summarized format that makes it easier to perform calculations of central tendency and to understand the overall shape of the data distribution.

Transcript

00:01 All right.

00:02 So for number five, we're given a list of numbers with frequencies, and we need to find the mean, the median, and the mode.

00:11 So the first thing we're going to focus on is getting the mean.

00:15 So the mean is accomplished by, first of all, adding up all of your frequencies in the frequency column.

00:22 So that's seven all the way down to the two.

00:24 So we do need to go and get that total.

00:32 So we're going to take seven plus.

00:35 15 plus 13 plus 16 plus 14 plus 15 plus 9 plus 2 and we should get 91 and then we are ready to go ahead and add this other column here where you multiply whatever numbers on the first of your list times the frequency number that goes with that so 12 times 7 for example so you're going to do 12 times 7 and you get 84, so go ahead and write 84.

01:11 11 times 15 is 165.

01:19 10 times 13 is 130.

01:25 9 times 16 is going to get us 144.

01:33 8 times 14 is 112.

01:39 7 times 15 is 105.

01:51 Okay, 6 times 9 is 54.

01:58 And five times two is 10.

02:01 So now that we've gotten all those multiplied out, we need to add up all those numbers.

02:07 Okay, so we're going to go ahead and do that here.

02:13 So we're going to add up 10 plus 54 plus 105, plus 112, plus 144, plus 130, plus 165, plus 84, plus 184, and you're going to get a total of 840 .4, so we're ready now to find the actual mean.

02:34 You're always going to take your big number divided by the little number.

02:37 So 804 divided by 91 is approximately 8 .8.

02:54 And depending on how your teacher wants you to round, it could be to the nearest hundredths place.

02:59 That just makes you pay attention to directions for that, but 8 .8 approximately.

03:04 And then now ready to find the median.

03:07 So the median is going to require us to get the cumulative frequency...

Please give Ace some feedback