Expected Value and Mean in Stats and Prob

6 min readJun 9, 2020

In this blog we will try to under stand what, how, why, when of Expected Value and Mean. so let’s start.

Mean :- it is used to find central tendency or middle value.
Expected Value :- it is used to find central tendency or middle value or improved version of mean.

You can see that definition of both concepts are same. Then what is difference between both. You can think of expected value as the advance or improved version of mean in simple term. How? Let’s explore it.

Mean = (X1+ X2 + X3+…. + Xn) / n

Let’s take an example to under stand the both concepts.

We know that in a dice there are 6 outcomes 1, 2, 3, 4, 5, 6. We role a dice and get a number it could be any number like 5 or 6 or 1 etc. To understand this concept let’s take a proper example. if we role a dice if it comes 1 it means we earn 1$ and if it is 2 then 2 $ and if it is 3 then we earn 3$ and so on. we know that that Probability of all outcomes is equal to 1/6 or uniform(same for all outcomes) keep in mind it is key point of the concept.

Now our task is to find how much dollars we can earn if we role the dice 1000 times?

If i role a dice 10 times the outcomes could be as follows 3526152436 or 1542252536 or 6154253421 or 2361553541 and so on. And when you find mean of these above value you will see that it is around 3.5( 6154253421, 6+1+5+4+2+5+3+4+2+1 / 10 = 3.3 ). You can take any outcomes(if it is taken randomly) and do the same calculation you will get around 3.5. if you want to get more accurate then increase the rolling time to 50 or more than that.

But what is 3.5. why we will get mean of the rolling dice(of any number of times) to 3.5.

Mean of outcome of dice = (1+2+3+4+5+6)/6 = 3.5

So 3.5 value says that if you role a dice 150 or 220 or 1000 or 10000 and so on(whatever rolling times but it should be larger) . We will get/earn 3.5 dollar/value per each role. For example if you role dice 100 times you will get around 100*3.5 = 350 dollar/value(it could be 340 or 370 or 335 and so on but it would be around 350). Note we can not judge it with less number of rolling like 10, 12 , 15 so on . For more accurate take larger number of rolling times. This method is known as law of large number(google it for more details).

So Answer of the above question, how much we can earn by rolling 1000 times would be around 1000*3.5 = 3500 dollar/ value.

So in conclusion we can say that mean tells us about central tendency or middle or average outcomes of an event .

Now what is the Expected Value how it is the same as mean and how we can say it it the improved version of mean. Let’s try to find out.

In above example if i do some magic or cheating with dice as follow the probability of 1 = 4/6, 2= 1/6, 3 = .25/6, 4= .25/6, 5= .25/6, 6 =.25/6. Previously it was 1/6 or uniform for all value/dollar. All probabilities sum is 4/6 + 1/6+ .25/6 + .25/6 + .25/6 + .25/6 = 1.

And then if we ask that how much dollar or value we can earn if we role dice 1000?

Would it be same as previous 3500$? Think about it.

Answer is wrong we can not get 3500$ if Probability of value has changed. Let’s try to understand why and if not than much we can earn.

Here we are giving less probability to 6 5 4 3 so that we are not able to earn much dollar. Probability of .25/6 show that if we role a dice 24 times then we will get 6 or 5 or 4 or 3 value/dollar . I just multiplied 4 with .25/ 4 = 1/24 to make .25 to 1. And if we compare this Probability with previous probability it was 1/6, means when we role a dice 6 times then we can get 1 time 6 or 5 or 4 or 3. which has more chances to earn dollar. Let’s come to another left dollar 1 and 2. If we see probability of 2 it is the same as previous so it won’t affect. Now 1 has high probability of coming 4/6 means if we role dice 6 times we will get 4 times 1 dollor. Means that it is giving less dollor to us.

The question is how much dollars we will earn if we role a dice 1000 times. Now expected value comes into picture in such above situation.

Expected Value E[ X ] = X1P(X1) + X2P(X2) +X3P(X3) ….. + XnP(Xn)

Where P(X) = probability of outcome X

According to above formula it will be as follow 1*4/6 + 2*1/6 + 3*.25/6 + 4*.25/6 + 5*.25/6 + 6*.25/6 = 1.75.

Then answer of how much earn if we role a dice 1000 times would be 1000*1.75 = 1750 dollar.

We can see that of we change the Probability then central tendency or middle will be change.

What we have observed from this example. Let’s examine it.

When we have equal or uniform Probability of outcomes then use mean to find central tendency or middle but if probability of outcomes change or not equal or non uniform then use expected value method to find central tendency or middle.

Let’s prove it mathematically. How mean and Expected Value is related to each other or improved version version of mean.

Mean = (X1 + X2+ X3 + X4+….+ Xn)/ n = 1/n(X1) + 1/n(X2) + 1/n(X3) + 1/n(X4) …. 1/n(Xn)

Expected Value E [ X ] = X1P(X1) + X2P(X2) +X3P(X3) + X4P(X4) …. Xn*P(Xn)

Mean = Expected Value

when P(X1) = P(X2) = P(X3)= P(X4) ….P(Xn) = 1/n

If we compare the formula of both then we can observe that if p(x1) = p(x2) = p(x3)= p(x4) ….p(xn) = 1/n then Expected value will be equal to mean.

When above condition p(x1) = p(x2) = p(x3)= p(x4) ….p(xn) = 1/n will look like this. Answer is when we will have Probability of all outcomes is equal. Let’s take couple of examples.

In our example we had first equal probability of outcomes 1/6 for all 1 2 3 4 5 6. Then we changed that Probability of outcomes to some other probabilities which was not equal or uniform. In that condition when we tried to find the central tendency or middle of earning that was wrong by using mean formula. Why, because probability of outcomes was not equal.

Let’s take 1 more example where we have to use expected value instead of mean. Suppose there is lottery going online. And there are 5 types of money we can get 0$, 5$, 20$, 50$, 100$. one ticket price for the lottery is 10$. Now if we ask that how much money we earn if we buy 100 tickets. If we use mean here then 0+5+20+50+100 / 5= 35$ for each ticket. By using this if we buy 100 then it would be 3500$ around. Great no loss. But do you think that it would be like this we will earn so much practically. Answer is definitely not. Why because in this there is no loss who bought the ticket but what about vendor they are in loss. We give them 100*10(each per ticket) = 1000 and in return we got 3500$ so your total net earning 2500$ but vendor has lost 2500$. Now the question is do you think it would be like so that vendor is in loss. Then how will vendors earns money. Here comes tricky part they(Vendors) put probability of coming 0$ highest, probability of 5 would have less probability, 20 will have less than 5 and 0, and 50 and 100 will have very very low probability. In this method when you use mean then answer would be wrong but using expected value would be great to find central tendency.

Conclusion:

Mean and Expected value both try to find out the central tendency of event when you take larger amount of event, it will be close to central value. But difference is that when Probability of outcomes is not equal or uniform then mean would give wrong result but expected value will give right result. So choose expected value when outcome has not equal probability other wise mean is ok. And we also compared that how expected value is nothing but improved version of mean mathematically.

I hope you have understood this concept. If you want to add some advice please comment below and don’t forget to clap.

Thank you 😊.