Certainly! Let's address each part of the question step-by-step:
### (a) What do you mean by Σf?
In statistics, Σf represents the sum of the frequencies (total number of observations) in a dataset. It is denoted as "n." Essentially, it indicates how many data points or observations are there in total.
### (b) Find the arithmetic mean (AM) of the given data.
To find the arithmetic mean (AM) of the data, you use the formula:
[tex]\[ \text{AM} = \frac{\Sigma fx}{n} \][/tex]
Given:
- Σfx (the sum of the product of frequencies and corresponding values) = 750
- n (number of observations) = 30
Plug these values into the formula:
[tex]\[ \text{AM} = \frac{750}{30} = 25.0 \][/tex]
### (c) To maintain the AM, what should be removed from Σf if 250 is subtracted from Σfx?
If 250 is subtracted from Σfx, and we want to maintain the same arithmetic mean of 25.0, we need to ensure that Σf remains the same since AM should not change.
New sum of products:
[tex]\[ \text{New Σfx} = 750 - 250 = 500 \][/tex]
To maintain the same AM, Σf should remain:
[tex]\[ \Sigma f = n = 30 \][/tex]
Thus, nothing should be removed from the total number of observations, which remains 30.
### (d) Find the changed percentage in AM if 10 observations were omitted in data.
If 10 observations are omitted, the new number of observations [tex]\( n' \)[/tex] is:
[tex]\[ n' = 30 - 10 = 20 \][/tex]
The original arithmetic mean calculated was 25.0. Now, we need to find the new arithmetic mean given that Σfx remains 750 but with only 20 observations:
[tex]\[ \text{New AM} = \frac{\Sigma fx}{n'} = \frac{750}{20} = 37.5 \][/tex]
To find the percentage change in AM:
[tex]\[ \text{Percentage Change} = \left( \frac{\text{New AM} - \text{Original AM}}{\text{Original AM}} \right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = \left( \frac{37.5 - 25.0}{25.0} \right) \times 100 = 50.0\% \][/tex]
So, the arithmetic mean increases by 50.0% when 10 observations are omitted.
