In 3-9, for a normal distribution, determine what percent of the data values are in each given r

In 3-9, for a normal distribution, determine what percent of...

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. Its bell-shaped curve is a common model for a wide range of natural phenomena.

Standard Deviation

Standard deviation is a measure of the dispersion or spread of a set of values around the mean in a data set. In the context of a normal distribution, intervals defined by multiples of the standard deviation (such as one or two standard deviations from the mean) correspond to specific percentages of the data.

Z-Score

A z-score is a standardized score that indicates how many standard deviations a data point is from the mean. It allows for comparing values from different normal distributions by converting them to a common scale, which is essential in finding probabilities and percentiles in the distribution.

Cumulative Distribution Function

The cumulative distribution function (CDF) of a normal distribution represents the probability that an observation will fall below a specific value. This function is key for calculating the percent of observations between any two points by finding the difference between their CDF values.

Empirical Rule

The empirical rule, sometimes called the 68-95-99.7 rule, is a guideline that states approximately 68% of values lie within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This rule provides a quick estimate of the spread and probability within a normal distribution.

Transcript

00:01 So you can see here that i've gone ahead and drawn out a normal distribution curve up here at the top of the screen.

00:05 I've drawn out two, but we're going to focus on the one on top to begin with here.

00:09 Gone ahead and drawn the mean there, marked the mean out, because the whole point of a normal distribution is that the mean should be right in the middle of everything.

00:18 And then i've also gone ahead and marked out one, two, and three standard deviations both above the mean, meaning to the right, and below the mean, meaning to the left.

00:28 Okay.

00:29 You will also notice off to the right here that i have gone ahead and written out what is called the empirical rule and what is covered in the notes in this section, which is that we know 68 % of our data is contained within one standard deviation from the mean, 95 % of our data is kept within two standard deviations of the mean, and 99 .7 % of our data is kept within three standard deviations of the mean.

00:53 As you guys can see what this question is about is asking us to figure out what percent of the number.

00:58 Data is contained within certain distance.

01:02 So i want to go ahead and break down each of these little pieces, each of these little slivers on the graph, and just go ahead and break out all those little percentages because then this problem will become much easier.

01:13 We're just going to, we'll just have to add a few percentages together.

01:16 So for example, 68 % of our data is kept within one standard deviation of the mean.

01:22 Well, that's both sides included.

01:24 So the whole point of a normal distribution is everything is being split, perfectly in half.

01:29 So 68 divided by 2 gives me 34, meaning this piece right here is 34%, and this piece right here is 34%.

01:40 That covers the total 68 % of the data.

01:45 So then if i want to find the second standard deviation, well i can't include the 68 % right.

01:51 We've already covered the 68%.

01:53 So while two standard deviations does hold 95 % of our data, what i can't include the 68%, what i'm i need to do is i need to take that 95 and go ahead and subtract out the 68 % to see what i actually have left to deal with.

02:08 If you take 95 minus 68, you have 27 % left over.

02:16 Again, we need to split that in half because that's what the whole point of a normal distribution curve is, is we are finding two standard deviations but on either side.

02:26 So 27 divided by two, gives us 13 .5%.

02:32 I ran out of screen there.

02:34 That's okay.

02:34 We're going to label it down here anyways.

02:36 It gives us 13 .5 % meaning this piece right here is 13 .5%.

02:42 And this piece right here is 13 .5%.

02:47 Finally, we know that three standard deviations holds 99 .7 % of our data...