I was reading this link – source – http://www.mathsisfun.com/data/standard-normal-distribution.html
Here they say that for the bell curve – the mean = mode = median are identical.
How mean i.e. average be equal to mode? What I understand is that median will be the point that is equidistant from the two extremes so here the peak in the bell curve will be the median.
Mode is the item that has occurred frequently – so the peak point has occurred twice – also , the extremes has occurred twice. but I take pairs (x, y)then probably I can say that the peak is the mode.
What I can't understand is the average? How the average will be the highest point. Basically, If I just consider three points – two extreme and the highest then the average will not be the highest point. So, I couldn't understand that the bell curve has the median, mode and mean equal. Can somebody please explain me this?
Best Answer
I guess that you are making a confusion between the values of the random variable (let X) and the frequencies.
We are not talking about the average height of the bars in the plot (i.e. the average frequency), we are talking about the average value of X, which is weighted by the frequency. The formula is the sum of X times the frequency of X over the sum of frequencies.
The mode is the value of X that achieves the highest frequency (i.e. the highest bar); it is not the frequency that occurs most often (often all frequencies are different).
The median is the value of X that reaches the half of the number of samples (you must accumulate the bars from left to right until you reach 50% of the total bar height); it is not "the bar in the middle".
The value of X which is the midpoint between the extremes is called the midrange.
In the case of a symmetric histogram, all these values are equal.