Question

The sum of all the values of a normalized histogram is equal to __________________ (in integer).

Hint:

When working with a normalized histogram, remember that it represents a probability distribution, and the sum of all the probabilities (bin values) must always be 1.

Answer 1

Correct Answer: 1

A histogram is a graphical representation of the distribution of a dataset. In a histogram, the data is divided into intervals, called bins, and the frequency (or count) of data points falling into each bin is plotted. When we create a normalized histogram, the frequency values of the bins are adjusted so that the total sum of the histogram equals 1. This means that the heights of the bars in the histogram represent probabilities, not just raw counts.
To understand this more clearly, let's break down the steps involved in normalizing a histogram: Step 1: Understanding the raw histogram.
In a raw histogram, each bar represents the frequency (or count) of data points in a specific bin. The height of the bar is simply the number of data points within that bin. If we have \( n \) bins, then the total area under the bars would be the total number of data points, denoted as \( N \). So, the sum of the raw histogram values would be \( N \), where: \[ \sum_{i=1}^{n} f_i = N \] where \( f_i \) represents the frequency of the \( i \)-th bin, and \( N \) is the total count of all the data points.
Step 2: Normalizing the histogram.
When we normalize the histogram, we divide each bin’s frequency by the total number of data points \( N \). This converts each frequency to a probability, which represents the relative likelihood of a data point falling into that bin. The normalized frequency for each bin is given by: \[ p_i = \frac{f_i}{N} \] where \( p_i \) is the normalized frequency (probability) of the \( i \)-th bin.
Step 3: Sum of the normalized histogram values.
Since the sum of the frequencies in the raw histogram is \( N \), the sum of the normalized frequencies is: \[ \sum_{i=1}^{n} p_i = \sum_{i=1}^{n} \frac{f_i}{N} = \frac{1}{N} \sum_{i=1}^{n} f_i = \frac{1}{N} \times N = 1 \] Thus, the sum of all the values of a normalized histogram equals 1. This is a fundamental property of probability distributions, where the total probability across all bins must sum to 1.
Final Answer: \[ \boxed{1} \]