The given values represent data for a sample. Find the variance and the standard deviation based

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

Key Concepts

Sample Standard Deviation

The sample standard deviation is the square root of the sample variance. It provides a measure of dispersion in the same units as the data, making it easier to understand the typical distance of observations from the mean. It is widely used to summarize the variability within a sample.

Sample Variance

Sample variance quantifies the spread of the data points around the arithmetic mean for a sample. It is calculated as the sum of the squared differences between each observation and the mean, divided by the degrees of freedom (n-1). This adjustment (n-1 rather than n) compensates for the bias in the estimation of the population variance.

Arithmetic Mean

The arithmetic mean is the sum of all the data values divided by the number of observations. It serves as the central or typical value of the data set, and is essential in further calculations like variance and standard deviation.

Frequency Distribution

A frequency distribution organizes data by displaying the number of times each value or range of values occurs within a data set. It is particularly useful when data is collected with repeated measurements or observations for the same value, and it simplifies the process of calculating statistical measures.

Sample Data

Sample data refers to a subset of observations taken from a larger population for the purpose of statistical analysis. In statistical computations, a sample is used to estimate properties or parameters of the overall population, such as the mean and variance.

Transcript

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

00:09 So first we have to know what this data is based on our table.

00:13 And we notice it's in frequency table format, which just means that the first column will be the occurrences or just the numbers in our data set.

00:22 And the second column here to the right will be the frequency in which they appear in our table.

00:27 So in this case, one will appear three times, two will appear three times, three will appear three times, and so on and so forth.

00:36 Notice that they all have the exact same frequency, so that actually makes the problem a little bit easier, and you actually don't even need to use the frequency formula because things will start canceling out if you do.

00:48 However, because first, this is a sample, and second, numbers can get mixed up if you forget to multiply or divide by something.

00:56 If you switch from the frequency formula to the general formula.

01:01 I myself actually did this by doing that, and then i messed up.

01:06 So i'm just going to go with the frequency formula because it's the most straightforward way, and it works.

01:15 And it's just like one extra step in the end.

01:18 So, yeah, let's get started.

01:20 I use it on a calculator to find this because even with numbers like these, it's still time -consuming, even though it's not that hard on your brain to do this step -by -step.

01:32 So i'm just going to explain each step that the calculator is taking.

01:37 So i hope you can follow.

01:40 All right.

01:41 So first is taking our two columns, the occurrences and the frequencies.

01:46 It's taking the total frequency, which means just the total number of numbers in our dataset, which is 21, because there's three of each thing, and there are seven things.

01:58 And now we're taking the frequency times our occurrence.

02:01 Doing so will enable us to find the total, the sum of everything in this data set, which is 84.

02:12 This is, the reason why it's 84 is because we have three ones, so we have to account for all three of them.

02:18 We have three twos, so we have to account for all three of those, and so on and so forth...

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