# Finding arithmetic mean, standard deviation, mode and median

On the market quality of the fruit was measured and following results came out:

Quality of Fruit( in measuring units ) 65 70 75 80 85 90 95 100
Number                                  2  3  2  5  8 7  5  3


Define:
a) arithmetic mean and standard deviation
b) mode and median

How to find what is wanted here?

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G o o g l e it! – iccthedral Jul 5 '12 at 21:13

There are $35$ items that have been assessed.

a) To find the mean, you need to calculate $$\tfrac{(2)(65)+(3)(70)+(2)(75)+(5)(80)+(8)(85)+(7)(90)+(5)(95)+(3)(100)}{35}.\tag{1}$$

The standard deviation would could be defined in a couple of different ways. I will use the one I guess is the one more likely for your course. For the sample variance, calculate first $$\tfrac{(2)(65^2)+(3)(70^2)+(2)(75^2)+(5)(80^2)+(8)(85^2)+(7)(90^2)+(5)(95^2)+(3)(100^2)}{35}.\tag{2}$$ Subtract the square of the sample mean calculated in $(1)$. That gives you the sample variance $s^2$. For the sample standard deviation, take the square root.

But perhaps in your course, the formula for the sample variance and standard deviation involves an $n-1$ instead of an $n$. In that case, you should multiply the $s^2$ that I described by $\frac{35}{34}$. Then for the sample standard deviation, take the square root as usual.

b) Since there are $35$ items, and the median is the "middle" number, count $18$ from the bottom, or $18$ from the top. We end up in the "$85$" slot, so the median is $85$.

The mode is the value that occurs most often. A quick scan shows that the value $85$ is the one.

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Excellent presentation! @André Nicolas . – Takarakaka Jul 5 '12 at 22:20

Hint.

Mean is the average, i.e. if I have $n$ numbers, $a_1, a_2,\dots, a_n$, their mean is $$\mu=\frac{a_1+a_2+\dots+a_n}{n}=\frac{\sum_{m=0}^n a_m}{n}$$

Standard deviation, $\sigma$, is given by

$$\sigma=\sqrt{\frac{\sum_{m=1}^n a^2_m}{n-1}-\frac{n}{n-1}\mu^2}$$

A mode is the number that most often appears in a set of data. For example, the mode of

1 2 2 3 4 5 6


is $2$.

Lastly, a median is the middle number of the data when it is sorted (or mean of the two middle numbers, if there is an even number of elements). The mode of the above numbers is $3$.

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But how to calculate modus since there are more results with two even numbers ( 2,2),(3,3),(5,5)? – Takarakaka Jul 5 '12 at 21:27
The mode is not necessarily unique, if that's what you mean. There could be many. – Argon Jul 5 '12 at 21:30