Range, variance, and standard deviation
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Range, variance, and standard deviation
Biostatistics
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Mean, median, and mode
Range, variance, and standard deviation
Standard error of the mean (Central limit theorem)
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Evan Debevec-McKenney
To understand a set of data - having a single number like the mean or median gives us a one number summary, but understanding how the data is distributed is also very important - and that’s where the range, variance, and standard deviation can be helpful.
For example, let’s say we are looking at the weight of 10 people and we divided them into groups A and B.
weight of group A(in kg) weight of group B(in kg)
40 45 50 55 60 10 30 50 70 90 .
Mean = (40+45+50+55+60)/5= 50kg Mean = (10+30+50+70+90)/5= 50kg
Now, if you calculate the mean weight of group A and group B, you will find both of them have the same value of 50 kg, but the weights of individuals in group A are much more centered around the mean than in group B.
So let’s start by looking at the range, which is the difference between the highest and lowest value in a dataset.
In group A, we have (60-40)=20 kg, whereas in group B we have (90-10)=80 kg.
So far so good. But now, let’s say we have decided to include another group called group C.
Weight of Group C (in kg) 10 45 50 55 90. Mean = (10+45+50+55+90)/5= 50 kg Range 90-10 = 80kg
So even when we change two data points, group C still has the same mean and range as group B since it depends only on the highest and lowest values, thus it provides no information about how the rest of the data points are distributed.
In this situation, it’s clear that we need a better idea of how all of the values are distributed and to do that we can look at the variance.
To calculate the variance, which is written out as σ2, we take each data point (x), subtract it from the mean (x-bar), and then we square this value so we don’t end up with a negative number.
Next, we add up the squared values and divide that result with the total number of data points (n).
So, let’s use this formula to calculate the variance for group A, B, and C where all three had a mean of 50.
So, for group A, we get: (40-50)2+(45-50)2+(50-50)2+(55-50)2+(60-50)2/(5)=50 kg2
For group B, we get: (10-50)2+(30-50)2+(50-50)2+(70-50)2+(90-50)2 /(5) = 800 kg2
Summary
Range, variance, and standard deviation are all measures of the spread or dispersion of a set of numerical data. The range is the difference between the highest and lowest values in the data set. Variance is a measure of how far each value in a set of data is from the mean value. Variance is calculated by taking the average of the squared differences of each value from the mean. The larger the variance, the more spread out the data is from the mean. Standard deviation is the square root of the variance and it measures the spread of data around the mean. The larger the standard deviation, the more spread out the data is from the mean.
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