The given values represent data for a sample. Find the...

The given values represent data for a sample. Find the variance and the standard deviation based on this sample. $\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 55 & {11} \\ {50} & {15} \\ {45} & {4} \\ {40} & {1} \\ {35} & {1} \\ {35} & {12} \\ {30} & {4} \\ \hline\end{array}$

The given values represent data for a sample. Find the variance and the standard deviation based on this sample.
$\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 55 & {11} \\ {50} & {15} \\ {45} & {4} \\ {40} & {1} \\ {35} & {1} \\ {35} & {12} \\ {30} & {4} \\ \hline\end{array}$

Key Concepts

Sample Standard Deviation

The sample standard deviation is the square root of the sample variance. It quantifies the amount of variation or dispersion in the sample data on the same scale as the original data, making it a widely used measure of how spread out the data points are around the sample mean.

Frequency Distribution

A frequency distribution is a table that displays the frequency of various outcomes in a sample. In statistics, it helps organize data into categories along with the count of observations in each category, making it easier to compute measures such as the mean, variance, and standard deviation when data is not provided as a simple list of numbers.

Sample Mean

The sample mean is the average value of the data points in a sample, computed as the sum of all observed values weighted by their frequency divided by the total number of observations. It serves as a measure of the central tendency of the data and is a critical component in calculating further statistics like variance.

Sample Variance

Sample variance is a measure of the spread or dispersion of the data points in a sample around the sample mean. It is calculated by finding the average of the squared differences between each data point and the sample mean, using n-1 in the denominator to correct for bias in the estimation of the population variance from a sample.

Transcript

00:01 Okay, so the question here is given our, given a sample set of data, which is in this table right here, we want to find the variance and the standard deviation.

00:13 Okay.

00:14 So first to interpret the set of data, first column is just what the numbers are in the data.

00:21 So we have 55, 50, 45s.

00:24 And then the second column is the frequency of which each one appears.

00:28 So 55 will appear 11 times, 50 will appear 15 times, 45 will appear four times, et cetera.

00:36 So in order to calculate the frequency and the standard, sorry, not the frequency, the variance and the standard deviation, because this step -by -step process is very tedious, even with a calculator, especially if we're doing with like large numbers like 55 and 50.

00:59 I actually went ahead and i used an online calculator for this.

01:05 Of course, you're welcome to do this as well if you're allowed to.

01:09 But what i'm going to do is because this online calculator shows each step of the way, i'm just going to explain what it's doing.

01:16 That way you can prop, that way from my explanation, you'll be able to do this step by step on your own.

01:23 Okay, let's get started.

01:26 All right.

01:28 So these first two columns are exactly the same as our data set.

01:32 It's just copying and pasting what we have from our data set.

01:37 Note that it's also finding the total number of numbers in our data set.

01:41 That's 61.

01:44 And now, okay, so what we do is we want to take the, in order to find the mean, first we take the frequency times our occurrence.

01:53 So like, for instance, 605 is a lot.