The table shows the number of correct answers on a test...

The table shows the number of correct answers on a test consisting of 15 questions. The table represents correct answers for a sample of the students who took the test. Find the standard deviation based on this sample. $\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Correct } & {6} & {7} & {8} & {9} & {10} & {11} & {12} & {13} & {14} & {15} \\ \hline \text { Frequency } & {2} & {1} & {3} & {3} & {5} & {8} & {8} & {5} & {4} & {1} \\ \hline\end{array}$

The table shows the number of correct answers on a test consisting of 15 questions. The table represents correct answers for a sample of the students who took the test. Find the standard deviation based on this sample.
$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Correct } & {6} & {7} & {8} & {9} & {10} & {11} & {12} & {13} & {14} & {15} \\ \hline \text { Frequency } & {2} & {1} & {3} & {3} & {5} & {8} & {8} & {5} & {4} & {1} \\ \hline\end{array}$

Key Concepts

Frequency Distribution

A frequency distribution is a summary of how often each value in a dataset occurs. It is used to quickly visualize the distribution of data points, particularly when there are many repeated values, as in test scores or survey responses.

Sample Mean

The sample mean is the arithmetic average calculated by summing all the data values and dividing by the number of data points. When the data is organized in a frequency distribution, each value is weighted by its frequency to determine the overall average, which serves as a measure of central tendency.

Sample Variance

The sample variance measures the average squared deviation of each data point from the sample mean, providing insight into the spread or dispersion of the sample data. It is calculated by summing the squared differences (weighted by frequency) and then dividing by the number of observations minus one, reflecting the sample estimation of variability.

Standard Deviation

Standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the original data. It indicates how much the data points tend to deviate from the mean, offering a clear sense of the spread of values in the data set.

Transcript

00:01 Okay, so the question here is to find the standard deviation of this set of data of students who took a test and this is the set of data that shows the number of correct answers they got on the test right here.

00:16 So in our first row we have the occurrences in this case, the correct answers, and in our second row we have the frequencies.

00:24 So just the number of times that number of correct answers appeared in this sample of data.

00:30 So like for instance, we have two occurrences.

00:34 Now, we two times six correct answers occurred, one time seven correct answers occurred, three times eight correct answers occurred, and so on and so forth.

00:45 So we want to find the standard deviation of this.

00:49 And because this process is very tedious to do on paper or step by step with a calculator, i actually went ahead, use an online calculator, got, what i needed.

01:03 But what i'm going to do is instead i'm going to explain this process that the calculator is using step by step.

01:09 That way it'll save us a lot of time, but i will still explain the process so you understand it.

01:15 Okay, let's get started.

01:17 So first we take our two rows and we put them into a table like so.

01:22 This is the first row right here and this is the second row.

01:26 And then what we do is first you want to find the mean.

01:29 So what we do is we take our first.

01:31 Frequency and we multiply by each number in our data set.

01:35 So there's two occurrences, i mean, two times, yes, two occurrences of six.

01:41 So it's 12.

01:42 One occurrence is seven, so one times seven to seven, and so on and so forth.

01:47 That's how we get these numbers in this column right here.

01:49 Then we add them up to get the total number of correct answers in this sample, which is 440.

01:56 We divide it by the total number of students or the total number of frequencies.

02:01 In this sample, which is 40, and we get that our average is exactly 11.

02:05 It's a nice number because it doesn't have any places after it's decimal.

02:10 It's good.

02:11 We can actually do this on paper without having to go with a lot of chaos with decimals...

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