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

The given values represent data for a population. Find the variance and the standard deviation for each set of data. $\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 2 & {0} \\ {4} & {1} \\ {6} & {6} \\ {8} & {6} \\ {8} & {10} \\ {10} & {13} \\ {12} & {21} \\ {12} & {7} \\ {16} & {1} \\ \hline\end{array}$

The given values represent data for a population. Find the variance and the standard deviation for each set of data.
$\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 2 & {0} \\ {4} & {1} \\ {6} & {6} \\ {8} & {6} \\ {8} & {10} \\ {10} & {13} \\ {12} & {21} \\ {12} & {7} \\ {16} & {1} \\ \hline\end{array}$

Key Concepts

Population Mean

The population mean is the weighted average of all data values in a data set. When the data is given in a frequency distribution, the mean is calculated by multiplying each data value by its frequency, summing these products, and then dividing by the total number of observations in the population. This central value is essential in variance and standard deviation calculations, as they measure how data points deviate from the mean.

Frequency Distribution

A frequency distribution organizes data into distinct values or intervals with their corresponding frequencies that show how many times each value occurs. This method of data presentation simplifies the process of statistical calculations, such as computing the mean and variance, by consolidating repeated data points into summarized groups.

Population Variance

Population variance is a measure of the dispersion of data points around the population mean. It is calculated by taking the weighted sum of squared differences between each data value and the mean, divided by the total number of observations. Variance provides insight into the spread or variability within the entire population.

Population Standard Deviation

The population standard deviation is the square root of the population variance, offering a measure of dispersion in the same units as the original data. It quantifies the average distance of data points from the mean, making it easier to interpret the variability in the data set.

Transcript

00:01 Okay, so the question here is to find the variance and the standard deviation of our set of data.

00:10 And in this question we can see here, this is our set of data in this little chart right here.

00:16 The first column shows the actual occurrence or like the set of data, the number in the set of data itself.

00:23 And the second column marks the frequency of which it occurs.

00:27 So just to understand this table a little bit, i'm just going to look at this table a little bit, i'm just going to look at it.

00:30 List out the first few elements of this data in like the format that we know regularly.

00:36 So there's no occurrences of two, so there's no twos, then four, then it would be one, four, and then this frequency of six is six, so we rate six, sixes, and then there's ten eights.

00:50 So we do this ten times, and et cetera.

00:57 I'm not going to do this whole thing.

00:58 It's a lot of stuff.

00:59 It's 59 to be exact.

01:01 So because finding the variance in the standard deviation would actually be incredibly time consuming to go step by step by paper or even like step by step with a calculator.

01:11 What i did was i actually use an online calculator.

01:15 I know it's a bit cheating, but for time purposes, this is probably the best way.

01:21 Although i will explain how like what each column of this table in the calculator represents because each of this is a step in the formula.

01:31 Just so that you guys, if you guys have to do a step by step, you guys know what to do.

01:35 Okay, so let's get started.

01:37 First, we have our list of occurrences right here.

01:40 This is just identical to the first column in our set of data, and our frequency f, this is identical to the second column in our set of data.

01:49 So now, first, if we're given frequencies, then what we do is we first multiply each frequency by each occurrence.

02:00 So basically, this 4 right here is just 4 times 1, the occurrence times the frequency.

02:06 This 36 is 6 times 6...

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