Show using an example the effect, if any, on the standard deviation of adding a data value to a set of data that is equivalent to the mean.
Not quite sure what "standard deviation of adding a data value to a set of data that is equivalent to the mean," means.
An example would also help... thanks.
Wait, what I'm understanding is that it's asking you just to add another data value that is the equal to the mean to your current set of data?
Also, the example and effect that it's asking to give can be shown by random numbers, right?
Ok, I think I got it now. So I have 5 numbers
(2, 3, 5, 8, and 13).
If the mean of these numbers is 6.2 (standard deviation is 3.77), based on the question I would then add the mean (6.2) to my 5 numbers (making it 6 numbers), and then recalculate the mean again, and also the standard deviation?
So the standard deviation after would be 3.44.
The answer would be that the deviation decreased?
You are correct. The standard deviation measures the variation from the mean. So if you add values that are close to or equal to the mean, that would decrease that variation or deviation from the mean.
Your example is a good one.
thank you :)
To understand the effect of adding a data value that is equivalent to the mean on the standard deviation, let's consider an example.
Let's say we have a set of data representing the weights of five apples: 100g, 150g, 200g, 250g, and 300g. The mean weight of these apples can be calculated by summing up all the weights and dividing by the number of data points, in this case, 5. So, the mean weight is (100+150+200+250+300)/5 = 200g.
The standard deviation measures the average amount by which each data point in a set deviates from the mean. It gives us a sense of the dispersion or spread of the data points. To calculate the standard deviation, we need to calculate the difference between each data point and the mean, square these differences, calculate the average, and finally take the square root.
In our example, the standard deviation of the apple weights before adding a data point equivalent to the mean (200g) would be:
Step 1: Calculate the difference between each data point and the mean:
(100-200)^2 = 10000
(150-200)^2 = 2500
(200-200)^2 = 0
(250-200)^2 = 2500
(300-200)^2 = 10000
Step 2: Calculate the average:
(10000+2500+0+2500+10000)/5 = 6000
Step 3: Take the square root:
sqrt(6000) ≈ 77.46
Therefore, the standard deviation before adding a data point equivalent to the mean is approximately 77.46g.
Now, let's add a data point of 200g (the mean) to our data set. So, the new set of data becomes: 100g, 150g, 200g, 200g, 250g, and 300g.
To calculate the standard deviation after adding the mean, we repeat the same steps as before:
Step 1: Calculate the difference between each data point and the mean:
(100-200)^2 = 10000
(150-200)^2 = 2500
(200-200)^2 = 0
(200-200)^2 = 0
(250-200)^2 = 2500
(300-200)^2 = 10000
Step 2: Calculate the average:
(10000+2500+0+0+2500+10000)/6 = 6666.67
Step 3: Take the square root:
sqrt(6666.67) ≈ 81.65
Therefore, the standard deviation after adding a data point equivalent to the mean is approximately 81.65g.
From this example, we can observe that adding a data value equivalent to the mean does not have a significant effect on the standard deviation. In this case, the standard deviation increased slightly from approximately 77.46g to approximately 81.65g. The exact change depends on the specific data set and the value added, but in general, adding a data value equivalent to the mean is unlikely to have a substantial impact on the standard deviation.