What does it mean by 'an' and 'n' in the equation?

[Indranil]

What does it mean by 'an' and 'n' in the equation below? and could you explain 'the sum up sign' in an easy way?

 

[lev888]

Looks like somebody forgot to finish editing the page before publishing.
Explanation: https://www.mathsisfun.com/algebra/sigma-notation.html

 

[Dr.Peterson]

What does it mean by 'an' and 'n' in the equation below? and could you explain 'the sum up sign' in an easy way?

It looks like what happens when you try to copy from some pages, and it gives you both the formatted and unformatted versions of the same thing.

Ignore the second copy of each of the three formatted bits (the list, and the two equations for "amean".)

 

[Indranil]

It looks like what happens when you try to copy from some pages, and it gives you both the formatted and unformatted versions of the same thing.

Ignore the second copy of each of the three formatted bits (the list, and the two equations for "amean".)

Still, I don't understand what 'n' and 'an'? could you simplify it, please?

 

[Denis]

1 2 3 4 ...........n
7 3 8 6 ...........4

a(3) = 8, a(n) = 4

 

[Dr.Peterson]

Still, I don't understand what 'n' and 'an'? could you simplify it, please?

Here is what it says:

Suppose the values obtained in several measurements are $\displaystyle a_1, a_2, a_3, ..., a_n$.
The arithmetic mean of these values is taken as the best possible value of the quantity under the given conditions of measurement as:
​

$\displaystyle \displaystyle a_{mean} = \frac{(a_1 + a_2 + a_3 + ... + a_n)}{n}$​

or​

$\displaystyle \displaystyle a_{mean} = \sum_{i=1}^{n} \frac{a_i}{n}$
​

n is the number of measurements. They are called a₁, a₂, and so on, up to the last (nth) one, a_n. That is, the subscript (1, 2, ..., n) distinguishes the first, second, ..., nth number.

The mean is the sum of all n measurements, divided by n, the number of measurements.

Is that clear?

 

[mmm4444bot]

Still, I don't understand [the meaning of] 'n' and 'an' …

Symbol a_n represents the last number in a sequence of measurements. (The sequence is named a.)

a = {a₁, a₂, a₃, … a_n}

a₁ is the 1st measurement in sequence a

a₂ is the 2nd measurement

a₃ is the 3rd measurment

The dots indicate that some measurements are not listed. (We don't write complete lists because they're too long. We show only the first few elements and the last one.)

a_n is the last measurement in the sequence (we call it the 'nth element' or 'nth term')

Therefore, symbol n represents the number of measurements.

The arithmetic mean is an average measurement. We calculate this average by adding all the measurements and dividing their total by the number of measurements.


$\displaystyle a_{\text{mean}} = \dfrac{a_1 + a_2 + a_3 + … + a_n}{n}$

$\displaystyle \displaystyle a_{\text{mean}} \; = \; \sum_{i=1}^{n} \bigg(\dfrac{a_i}{n} \bigg) \;$$\displaystyle \displaystyle = \; \dfrac{1}{n} \cdot \sum_{i=1}^{n} \bigg( a_i \bigg)$

Note the additional part (shown in green). The summation in black shows each measurement being divided by n (then those fractions are added). But, there is a property of summations that allows us to factor out 1/n, so we can add the elements first and then divide by n once (at the end). :cool:

 

[Indranil]

Symbol a_n represents the last number in a sequence of measurements. (The sequence is named a.)

a = {a₁, a₂, a₃, … a_n}

a₁ is the 1st measurement in sequence a

a₂ is the 2nd measurement

a₃ is the 3rd measurment

The dots indicate that some measurements are not listed. (We don't write complete lists because they're too long. We show only the first few elements and the last one.)

a_n is the last measurement in the sequence (we call it the 'nth element' or 'nth term')

Therefore, symbol n represents the number of measurements.

The arithmetic mean is an average measurement. We calculate this average by adding all the measurements and dividing their total by the number of measurements.


$\displaystyle a_{\text{mean}} = \dfrac{a_1 + a_2 + a_3 + … + a_n}{n}$

$\displaystyle \displaystyle a_{\text{mean}} \; = \; \sum_{i=1}^{n} \bigg(\dfrac{a_i}{n} \bigg) \;$$\displaystyle \displaystyle = \; \dfrac{1}{n} \cdot \sum_{i=1}^{n} \bigg( a_i \bigg)$

Note the additional part (shown in green). The summation in black shows each measurement being divided by n (then those fractions are added). But, there is a property of summations that allows us to factor out 1/n, so we can add the elements first and then divide by n once (at the end). :cool:

Thanks a lot for all of your kind efforts for me to learn math in an easy way. Thank you all again.

 

[Indranil]

How to learn summation in an easy way because it gives me a headache?

 

[mmm4444bot]

Study lots of examples, and practice. :cool: Google kewords like summation lessons examples, and look for resources that you find easy to understand. If one page is too hard, try another.

Here are two search results; the first is a short video introduction, and the second is a PDF file containing several examples.

https://www.khanacademy.org/math/al...ries/alg2-sigma-notation/v/sigma-notation-sum

http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-sigma-2009-1.pdf

 

[Indranil]

The arithmetic mean is an average measurement. We calculate this average by adding all the measurements and dividing their total by the number of measurements.


$\displaystyle a_{\text{mean}} = \dfrac{a_1 + a_2 + a_3 + … + a_n}{n}$

$\displaystyle \displaystyle a_{\text{mean}} \; = \; \sum_{i=1}^{n} \bigg(\dfrac{a_i}{n} \bigg) \;$$\displaystyle \displaystyle = \; \dfrac{1}{n} \cdot \sum_{i=1}^{n} \bigg( a_i \bigg)$

Note the additional part (shown in green). The summation in black shows each measurement being divided by n (then those fractions are added). But, there is a property of summations that allows us to factor out 1/n, so we can add the elements first and then divide by n once (at the end). :cool:

It would be very helpful for me if you kindly explain the summation above in an easy way.

 

[Dr.Peterson]

It would be very helpful for me if you kindly explain the summation above in an easy way.

It will be very helpful for us if you could tell us what part of what has been explained is not "easy" for you. You might tell us what you think it means, in your own words, so we can see how much you understand, and also tell us specifically which parts you are unsure of.

Keep in mind that mathematics is not easy, and nothing anyone does can make it not require mental effort. But we can make that effort easier for you if we know how you think, and can adjust what we say to that. What is "easier" depends on the person. This is why specific questions are much more likely to get good answers.

 

[mmm4444bot]

… kindly explain the summation above in an easy way.

Which part do you need help understanding, in the Khan Academy introduction video? Please post the timestamp and your specific question. :cool:

 

[Indranil]

$\displaystyle a_{\text{m}\text{ean}} = \dfrac{a_1 + a_2 + a_3 + … + a_n}{n}$

$\displaystyle \displaystyle a_{\text{m}\text{ean}} \; = \; \sum_{i=1}^{n} \bigg(\dfrac{a_i}{n} \bigg) \;$$\displaystyle \displaystyle = \; \dfrac{1}{n} \cdot \sum_{i=1}^{n} \bigg( a_i \bigg)$

What does it mean by '1/n', 'n' on the top of the summation sign and 'i=1' on the bottom of the summation sign and 'ai' in the bracket?

 

[Denis]

No chalk/blackboard here; have a look here:

http://onlinestatbook.com/2/introduction/summation.html

If you'd like more sites, google "introduction to summation formula".

 

[JeffM]

What does it mean by '1/n', 'n' on the top of the summation sign and 'i=1' on the bottom of the summation sign and 'ai' in the bracket?

Are you seriously unsure of what $\displaystyle \dfrac{1}{n}$ means in mathematics?

If so, look up fraction by clicking on the word in blue.

$\displaystyle \Sigma$ is the capital Greek letter sigma, equivalent to S in English, and stands for "sum." ($\displaystyle \Pi$ is the capital Greek letter pi, equivalent to P, and stands for "product.")

$\displaystyle \displaystyle \sum_{j=k}^n$ means that we are to sum (add) n expressions, each of which will contain the variable j, with j starting at k and increasing by 1 each time.

$\displaystyle \displaystyle \sum_{j=5}^8 \left ( 2^j * x_{j-4} \right ) =$

$\displaystyle 2^5 * x_{5-4} + 2^6 * x_{6-4} + 2^7 * x_{7-4} + 2^8 * x_{8-4} = 32x_1 + 64x_2 + 128x_3 + 256x_4.$

It is a very compact notation if n is unknown or if n is large.

$\displaystyle n = 3 \implies \displaystyle \dfrac{1}{n} * \sum_{j=1}^n \left ( x_j \right ) = \dfrac{1}{3} * (x_1 + x_2 + x_3) = \dfrac{x_1 + x_2 + x_3}{3}$

If n = 500, writing it out the latter way is inefficient.

 

[mmm4444bot]

What does it mean by '1/n', 'n' on the top of the summation sign and 'i=1' on the bottom of the summation sign and 'ai' in the bracket?

n represents how many numbers are added. If we add three numbers, then n=3. If we add 4,527 numbers, then n=4527.

Multiplication by 1/n is the same as division by n. Therefore, the factor 1/n shows that we are dividing the total by n.

i represents the index. The index counts the numbers and shows their position in the set. We start counting at 1 (the first number in set a), and the notation i=1 shows this.

a_i is a symbol that represents each number being added. When i=1, symbol a_i becomes a₁, and a₁ represents the first number being added. When i=2, the symbol a_i becomes a₂, and it represents the second number being added. The index continues counting each number, until it reaches the number n. At that point, we have symbol a_n, and a_n represents the last number being added (often called the "nth number").

Let's add the three numbers in this set: a = {33, 62, 85}

33 + 62 + 85

We could also use symbols to represent these numbers.

The first number is 33, so let's call it a₁
The second number is 62, so let's call it a₂
The third number is 85, so let's call it a₃

Therefore a₁ + a₂ + a₃ means 33 + 62 + 85

The subscripts (in red) show us the index i counting each number (first number, second number, third number). In other words, the index starts at i=1 and it ends at i=3.

In this example, n=3 because we're adding three numbers. The nth number is a₃

What if we wanted to add 3,000 numbers, instead? Nobody wants to write 3,000 numbers, so we need a shorter way to write sums. Sigma notation gives us a shorter way.

Using my set {33, 62, 85} we understand that a₁=33, a₂=62, and a₃=85. We write their sum easily using Sigma notation, like this:

$\displaystyle \displaystyle \sum_{i=1}^{3} \bigg( a_i \bigg)$

You can see the count starts at i=1. You can also see that n=3 (this is what tells the reader that we're adding three numbers). You can see the symbol representing numbers being added (a_i).

i=1 tells us to start the sum with the first number a₁

33 +

Next, the index increases from i=1 to i=2. Sigma notation tells us to add the next number a₂

33 + 62 +

Next, the index increases from i=2 to i=3. Sigma notation tells us to add the next number a₃

33 + 62 + 85

At this point, the index has reached n, so all numbers have been added.

$\displaystyle \displaystyle \sum_{i=1}^{3} \bigg( a_i \bigg) = 33 + 62 + 85 = 180$

If we want to find the average (m͏ean) of set a={33, 62, 85}, then we divide the total by the count.

180/3 = 60

The mean is 60. Dividing by 3 is the same as multiplying by 1/3. Therefore, we could also write the calculation like this:

1/3 ∙ (33 + 62 + 85)

This is why the Sigma notation for the mean of a_i looks like this:

$\displaystyle \displaystyle a_{\text{mean}} = \dfrac{1}{3} \cdot \sum_{i=1}^{3} \bigg( a_i \bigg)$


We could also write $\displaystyle \displaystyle \dfrac{\sum_{i=1}^{3} (a_i)}{3}$ but most people write it the first way (multiplying by 1/n instead of writing ratio form).


Here is a final example:

$\displaystyle \displaystyle \sum_{i=1}^{208} \bigg( x_i \bigg)$

I can tell by looking at this notation that we are adding 208 numbers in a set called x, and subscripted symbol x_i is a generic variable used to represent the numbers. When index i starts counting (1,2,3,…), then x_i represents the individual numbers in set x added one by one (in order of listing).

The Sigma notation is much easier than writing the sum like this:

5+35+74+53+29+72+17+82+98+43+183+71+163+8+31+63+32+1+53+73+13+28+31+52+158+3+41+5+532+223+75+52+651+987+451+121+851+451+151+625+403+322+358+233+453+532+305+23+342+145+472+312+544+293+454+1023+453+405+984+268+234+957+2034+9670+9345+7093+4867+5394+750+923+4709+3750+9232+5703+8975+6039+5730+9572+3095+7093+857+9325+7034+9793+4754+385+345+832+570+934+759+35+7+1389+757+235+74+97+54+3+95+72+34+9+5+35+74+53+29+72+17+82+98+43+183+71+163+8+31+63+32+1+53+73+13+28+31+52+158+3+41+5+532+223+75+52+651+987+451+121+851+451+151+625+403+322+358+233+453+532+305+23+342+145+472+312+544+293+454+1023+453+405+984+268+234+957+2034+9670+9345+7093+4867+5394+750+923+4709+3750+9232+5703+8975+6039+5730+9572+3095+7093+857+9325+7034+9793+4754+385+345+832+570+934+759+35+7+1389+757+235+74+97+54+3+95+72+34+9 :cool: