i have a question regarding statistics. i just don't know how to do it and would really appreciate if someone could tell how i can solve it.
the question is;
a) compute the mean and the standard deviation of the population?
ive looked through all my notes and my textbook and i just cant figure out how to this.
thank you :)
A population is a collection of data, and a sample is a subset (i.e. a smaller number of the data, obtained randomly) of the population.
The population mean, μ, is the arithmetic average of ALL the data present in the population. The sample mean is the arithmetic mean of the particular sample. It can vary from sample to sample, but is generally close to the population mean.
For example:
Consider a class of 10 children with ages 8,8,9,9,9,10,10,8,9,10.
The population mean is
μ=(8+8+9+9+9+10+10+8+9+10)/10
=90/10
=9.0
If we take a random sample, say, every second child starting from the first, we get
sample mean
=(8+9+9+10+9)
=45/5
=9.0
Which happens to be identical to the population mean. This is why often we take the sample mean to represent the population mean.
The standard deviation requires a little more calculation, it is defined, for the population, as:
σ
=√((Σ(x-μ)²)/n)
In the above case, we have
σ
=√(((8-9)²+(8-9)²+(9-9)²+(9-9)²+(9-9)²++(10-9)²+(10-9)²+(8-9)²+(9-9)²+(10-9)²)/10)
=√((1²+1²+0²+0²+0²+1²+1²+1²+0²+1²)/10)
=√(6/10)
=0.7746...
The sample standard deviation, sn is calculated similarly, with the exception that the divisor n is usually replaced by (n-1) to give a better estimate of the population standard deviation.
To compute the mean and standard deviation of a population, you follow these steps:
1. Collect your data: Gather all the values in your population. For example, if you are studying the heights of a population of individuals, you would collect the heights of all individuals in your study.
2. Calculate the mean (average): Add up all the values and then divide by the total number of values. The formula for the mean is:
Mean = Sum of all values / Total number of values
3. Find the difference between each value and the mean: Subtract the mean from each value in the population.
4. Square each difference: Square each difference to eliminate negative values.
5. Calculate the sum of the squared differences: Add up all the squared differences obtained in the previous step.
6. Divide the sum of squared differences by the total number of values: This will give you the variance.
7. Find the square root of the variance: This will give you the standard deviation.
To summarize, the steps are:
1. Collect your data.
2. Calculate the mean.
3. Find the difference between each value and the mean.
4. Square each difference.
5. Calculate the sum of the squared differences.
6. Divide the sum of squared differences by the total number of values to get the variance.
7. Take the square root of the variance to get the standard deviation.
Note: If you have access to statistical software like Excel, SPSS, or Python, you can use these tools to perform the calculations automatically.