# histogram normalised to area 1

i have a histogram with the y-axis showing the proportion in percentage. That makes sense to me but now i have read that histograms can be normalized with the result that the area of the rectangles is 1.

Can someone explain me the idea,reason behind this? Many thanks!

## 1 Answer

Normalized histogram is on the same scale as density, and this is convenient if you want to compare empirical histograms with some theoretically obtained density (i.e. to superpose them on the same graph), or compare two histograms with differently selected bins.

Added: We can view each bar in the normalized histogram as an estimate of the value of probability density function at some nearby point (provided that density doesn't change much along the width of each bin). If two histograms come from the same distribution, both should resemble the same density function --- up to uncertainty introduced by sampling. Without normalization, the height of each bar would also depend on the sample size and the width of each bin.

Note that there are, also, other ways to visually compare empirical distributions of two random samples, for example Q-Q plots.

• thx zoran. You have also mentioned "compare two histograms with differently selected bins". Does this mean even the histograms have a different x-axis or y-axis, as long the area is 1 u can superpose(= look vor overlaps?) them? – mister nobody May 8 '14 at 13:47
• I would put it this way: in order to meaningfully compare two histograms on the same graph, you need to normalize them both. I expanded a little bit my answer above. – Zoran Loncarevic May 9 '14 at 11:46