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Sampling variation and boxplots Hi, I'm Dr Nic
In this video I am going to talk about sampling variation and sample size using box plots.
If you are not sure what a box plot, or box and whisker plot is,
you need to watch our video about it. The link is in the description below.
It might also be useful to watch the video Sources of variation
For this video we are going to use a pretend population of red apples.
For this pretend or simulated population, the weights are modelled with a normal distribution
with a mean weight of 100g and a standard deviation of 20g. The median is also 100 g.
Note that in real life we do NOT generally know the mean
and standard deviation of the whole population.
Say our population was all the apples in a large orchard.
We could not practically weigh each individual apple. We will take one sample of 20 apples.
We draw a boxplot showing the minimum and maximum weights, the median
and the upper and lower quartiles. The box tells us where the middle 50% of the sample lies.
This gives us an idea of the distribution of the apple weights in this sample.
Again in real life we would only take one sample, but in this video we are exploring
what different samples would look like, so we will take many samples from this same population.
Here you can see nine samples of 20 apples each, summarised as boxplots.
There is quite a lot of variation.
Remember - every single one of those samples came from the same population.
You would only take one sample, and that is what you would use to draw conclusions or inference about the population of all the apples.
You can see that the medians range from about 85 to about 110.
The medians are around 100 g, which is the population median.
Samples 5 and 6, barely overlap their boxes which hold the middle 50% of the sample,
but the other samples have considerable overlap.
Here are some more sets of samples of 20 apples each.
These are all taken at random from the same population.
Most of the time the boxes overlap.
This is important to think about, as we want to be able to look at two samples
and decide if they could come from the same population or not.
Now let's look at the effect of sample size.
What if we only had a sample of ten apples. What would you expect the boxplots to look like?
Would the sample distributions shown by the boxplots vary more than, less than or the same
as the samples of twenty apples? Pause and think about this and decide.
Here are some more sets of samples, this time of ten apples each, all taken from the same population as before.
You can see that there is more variation in these samples.
The medians are still around 100 g, but have more variation, and the boxes vary considerably in width.
You can see this with a set of samples of size 10 and a set of samples of size 20.
So now lets look at samples of 30 apples.
What would you expect the boxplots to look like?
Would the sample distributions shown by the boxplots vary more than, less than or the same as the samples of twenty apples?
As you can see, with a bigger sample size, the samples are more similar to each other.
A larger sample reduces the effect of sampling variation.
Finally, we will look at samples of size 50.
What do you think will happen?
Here are some sets of samples, this time of fifty apples each, all taken from the same population.
We can see that the larger samples are more similar to each other, still.
Do you notice anything else about the larger samples?
They tend to have a larger range from minimum to maximum than the smaller samples,
as there is more opportunity for extreme values to appear.
Here's a question for you.
Have a look at this set of samples
What sample size do you think these samples came from?
If you guessed smaller than ten, you are correct. These were samples of just 5 apples.
You would not expect samples of 5 apples from an orchard to give you a consistent distribution.
You may be wondering what the population boxplot would look like.
In this illustration we have modelled the weight distribution of the apples as normal with a mean
of 100 and a standard deviation of 20. This would give a box plot looking like this.
The distributions of the samples of size 30 were pretty close to the population.
When we come to compare two boxplots to see if the samples came from different populations,
it is worthwhile to think about how different or similar the samples are likely to be coming from the same population.
We will explore this more in another video, "Comparing two samples, using boxplots."
In the description below are links to further videos around the statistical concepts covered in this video.
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