What happens to the mean when you add or subtract a number from each data value?
Video transcript- "Ana played five rounds of golf "and her lowest score was an 80. " The scores of the first four rounds and the lowest round "are shown in the following dot plot." And we see it right over here. The lowest round she scores an 80, she also scores a 90 once, a 92 once, a 94 once, and a 96 once. "It was discovered that Ana broke some rules when she scored "80, so that score", so I guess cheating didn't help her, "so that score will be removed from the data set." So they removed that 80 right over there. We're just left with the scores from the other four rounds. "How will the removal of the lowest round "affect the mean and the median?" So let's actually think about the median first. So the median is the middle number. So over here when you had five data points the middle data point is gonna be the one that has two to the left and two to the right. So the median up here is going to be 92. The median up there is 92. And what's the median once you remove this? Now you only have four data points. When you're trying to find the median of an even number of numbers you look at the middle two numbers. So that's a 92 and a 94. And then you take the average of them. You go halfway between them to figure out the median. So the median here is going to be, let me do that a little bit clearer. The median over here is going to be halfway between 92 and 94 which is 93. So the median, the median is 93. Median is 93. So removing the lowest data point in this case increased the median. So the median, let me write it down here. So the median increased by a little bit. The median increases. Now what's going to happen to the mean? What's going to happen to the mean? Well one way to think about it without having to do any calculations is if you remove a number that is lower than the mean, lower than the existing mean, and I haven't calculated what the existing mean is, but if you remove that the mean is going to go up. The mean is going to go up. So hopefully that gives you some intuition. If you removed a number that's larger than the mean your mean is, your mean is going to go down cause you don't have that large number anymore. If you remove a number that's lower than the mean, well you take that out, you don't have that small number bringing the average down and so the mean will go up. But let's verify it mathematically. So let's calculate the mean over here. So we're gonna add 80, plus 90, plus 92, plus 94, plus 96. Those are our data points. And that gets us: two plus four is six, plus six is 12. And then we have one plus eight is nine, and this is, so these are nine and then you have another nine, another nine, another nine, another nine. You essentially have, this is five nines right over here. So this is going to be 452. So that's the sum of the scores of these five rounds, and then you divide it by the number of rounds you have. So it would be 452 divided by five. So 452 divided by five is going to give us, five goes into, it doesn't go into four, it goes into 45 nine times. Nine times five is 45, you subtract, get zero, bring down the two. Five goes into two zero times, zero times five is, zero times five is zero, subtract. You have two left over, so you can say that the mean here, the mean here is 90 and 2/5. Not nine and 2/5, 90 and 2/5. So the mean is right around here. So that's the mean of these data points right over there. And if you remove it what is the mean going to be? So here we're just going to take our 90, plus our 92, plus our 94, plus our 96, add 'em together. So let's see, two plus four plus six is 12. And then you add these together you're gonna get 37. 372 divided by four, cause I have four data points now, not five. Four goes into, let me do this in a place where you can see it. So four goes into 372, goes into 37 nine times. Nine times four is 36, subtract, you get a one. Bring down the two, it goes exactly three times. Three times four is 12. You have no remainder. So the median and the mean here are both, so this is also the mean. The mean here is also 93. So you see that the median, the median went from 92 to 93, it increased. The mean went from 90 and 2/5 to 93. So the mean increased by more than the median. They both increased but the mean increased by more. And it makes sense cause this number was way, way below all of these over here. So you could imagine if you take this out the mean should increase by a good amount. But let's see which of these choices are what we just described. "Both the mean and the median will decrease", nope. "Both the mean and the median will decrease", nope. "Both the mean and the median will increase, "but the mean will increase by more than the median." That's exactly, that's exactly, what happened. The mean went from 90 and 2/5 or 90.4, went from 90.4 or 90 and 2/5 to 93. And then the median only increased by one. So this is the right answer. Show
Standard deviation is used in statistics to tell us how “spread out” the data points are. Having one or more data points far away from the mean indicates a large spread – but there are other factors to consider. So, what affects standard deviation? Sample size, mean, and data values affect standard deviation, since they are used to calculate standard deviation. Removing outliers changes sample size and may change the mean and affect standard deviation. Multiplication and changing units will also affect standard deviation, but addition will not. Of course, it is possible by chance that removing an outlier will leave the standard deviation unchanged. It is important to go through the calculations to see exactly what will happen with the data. In this article, we’ll talk about the factors that affect standard deviation (and which ones don’t). We’ll also look at some examples to make things clear. Let’s get started. (You can also see a video summary version of this article on YouTube!) Standard deviation has the formula Standard deviation is used in fields from business and finance to medicine and manufacturing. Some of the things that affect standard deviation include:
Let’s take a look at each of these factors, along with some examples, to see how they affect standard deviation. Does Sample Size Affect Standard Deviation?Sample size does affect the sample standard deviation. However, it does not affect the population standard deviation. The sample size, N, appears in the denominator under the radical in the formula for standard deviation.  So, changing the value of N affects the sample standard deviation. Changing the sample size N also affects the sample mean (but not the population mean). Example 1: Changing N Changes Standard DeviationFor the data set S = {1, 3, 5}, we have the following:
If we change the sample size by removing the third data point (5), we have:
So, changing N changed both the mean and standard deviation. Of course, it is possible by chance that changing the sample size will leave the standard deviation unchanged. Example 2: Changing N Leaves Standard Deviation UnchangedFor the data set S = {1, 2, 2.36604}, we have the following:
If we change the sample size by removing the third data point (2.36604), we have:
So, changing N lead to a change in the mean, but leaves the standard deviation the same. Does Removing An Outlier Affect Standard Deviation?Removing an outlier affects standard deviation. In removing an outlier, we are changing the sample size N, the mean, and thus the standard deviation. Example: Removing An Outlier Changes Standard DeviationFor the data set S = {1, 3, 98}, we have the following:
If we change the sample size by removing the third data point (98), we have:
So, changing N changed both the mean and standard deviation (both in a significant way). Does Addition Affect Standard Deviation?Addition of the same value to every data point does not affect standard deviation. However, it does affect the mean. This is because standard deviation measures how spread out the data points are. Adding the same value to every data point may give us larger values, but they are still spread out in the exact same way (in other words, the distance between data points has not changed at all!) Example: Addition Does Not Change Standard Deviation.For the data set S = {1, 2, 3}, we have the following:
If we add the same value of 5 to each data point, we have:
So, adding 5 to all data points changed the mean (an increase of 5), but left the standard deviation unchanged (it is still 1). Does Multiplication Affect Standard Deviation?Multiplication affects standard deviation by a scaling factor. If we multiply every data point by a constant K, then the standard deviation is multiplied by the same factor K. In fact, the mean is also scaled by the same factor K. Example: Multiplication Scales Standard Deviation By A Factor Of KFor the data set S = {1, 2, 3}, we have the following:
If we use multiplication by a factor of K = 4 on every point in the data set, we have:
So, multiplying by K = 4 also multiplied the mean by 4 (it went from 2 to 8) and multiplied standard deviation by 4 (it went from 1 to 4). Does Changing Units Affect Standard Deviation?Changing units affects standard deviation. Any change in units will involve multiplication by a constant K, so the standard deviation (and the mean) will also be scaled by K. Example: Changing Units Changes Standard DeviationFor the data set S = {1, 2, 3} (units in feet), we have the following:
If we want to convert units from feet to inches, we use multiplication by a factor of K = 12 on every point in the data set, we have:
So, multiplying by K = 12 also multiplied the mean by 12 (it went from 2 to 24) and multiplied standard deviation by 12 (it went from 1 to 12). Does Mean Affect Standard Deviation?Mean affects standard deviation. To calculate standard deviation, we add up the squared differences of every data point and the mean. However, it can happen by chance that a different mean will lead to the same standard deviation (for example, when we add the same value to every data point). This is because standard deviation measures how far each data point is from the mean. So, the data set {1, 3, 5} has the same standard deviation as the set {2, 4, 6} (all we did was add 1 to each data point in the first set to get the second set). See the example from earlier (adding 5 to every data point in the set {1, 2, 3}): the mean changes, but the standard deviation does not. You can learn more about the difference between mean and standard deviation in my article here. ConclusionNow you know what affects standard deviation and what to consider about outliers and sample size. You can learn about how to use Excel to calculate standard deviation in this article. You can learn about the units for standard deviation here. You can learn about the difference between standard deviation and standard error here. You might also be interested to learn more about variance in my article here. You can learn more about standard deviation calculations in this resource from Texas A&M University. Also, Penn State University has an article on how standard deviation can be used to measure the risk of a stock portfolio, based on variability of returns. This article I wrote will reveal what standard deviation can tell us about a data set. I hope you found this article helpful. If so, please share it with someone who can use the information. Don’t forget to subscribe to my YouTube channel & get updates on new math videos! ~Jonathon What happens to the standard deviation when you add or subtract a number from each data value?Addition – adding (or subtracting) the same value to every data point will change the mean, but it will not change the standard deviation. Changing Units – changing units is multiplication by a constant, so it will affect standard deviation (for example, changing from feet to inches means multiplying by 12).
What happens to the mean and standard deviation of a number is added to each data value?Thus s'=s . As a general rule, the median, mean, and quartiles will be changed by adding a constant to each value. However, the range, interquartile range, standard deviation and variance will remain the same.
How does the mean change when the same number is added to or subtracted from each score in a data set?Adding, subtracting, multiplying, or dividing each score by a constant: When every score in a distribution is changed by the same constant, the mean will change by that constant. For example, if we add a constant of 5 to each score in a distribution, then the mean will increase by 5.
What happens to the mean when you add a constant?If you add a constant to every value, the mean and median increase by the same constant. For example, suppose you have a set of scores with a mean equal to 5 and a median equal to 6. If you add 10 to every score, the new mean will be 5 + 10 = 15; and the new median will be 6 + 10 = 16.
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