Problem 5 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Suppose we construct a histogram with bins \([0,1],(1,3],(3,5],(5,8]\), \((8,11],(11,14]\), and \((14,18]\). Given are the values of the empirical distribution function at the boundaries of the bins: \begin{tabular}{ccccccccc} \(t\) & 0 & 1 & 3 & 5 & 8 & 11 & 14 & 18 \\ \hline\(F_{n}(t)\) & 0 & \(0.225\) & \(0.445\) & \(0.615\) & \(0.735\) & \(0.805\) & \(0.910\) & \(1.000\) \end{tabular} Compute the height of the histogram on each bin.

Step-by-Step Solution

Answer

The histogram heights are 0.225, 0.11, 0.085, 0.04, 0.0233, 0.035, 0.0225.

Step 1: Identify Bins and Empirical Distribution Values

Given the bins are \([0,1], (1,3], (3,5], (5,8], (8,11], (11,14], (14,18]\), and the values of the empirical distribution function \(F_n(t)\) at the specified \(t\) values are provided. These values represent cumulative frequency at each boundary.

Step 2: Calculate Bin Widths

The width of each bin is the difference between its upper and lower boundary. Calculate:- \( [0,1] \): width = \( 1 - 0 = 1 \)- \( (1,3] \): width = \( 3 - 1 = 2 \)- \( (3,5] \): width = \( 5 - 3 = 2 \)- \( (5,8] \): width = \( 8 - 5 = 3 \)- \( (8,11] \): width = \( 11 - 8 = 3 \)- \( (11,14] \): width = \( 14 - 11 = 3 \)- \( (14,18] \): width = \( 18 - 14 = 4 \)

Step 3: Calculate Cumulative Count Differences for Each Bin

Use the empirical distribution values to find the cumulative count difference for each bin:- \( [0,1] \): difference = \( 0.225 - 0 = 0.225 \)- \( (1,3] \): difference = \( 0.445 - 0.225 = 0.22 \)- \( (3,5] \): difference = \( 0.615 - 0.445 = 0.17 \)- \( (5,8] \): difference = \( 0.735 - 0.615 = 0.12 \)- \( (8,11] \): difference = \( 0.805 - 0.735 = 0.07 \)- \( (11,14] \): difference = \( 0.910 - 0.805 = 0.105 \)- \( (14,18] \): difference = \( 1.000 - 0.910 = 0.09 \)

Step 4: Calculate Histogram Height for Each Bin

The height of the histogram for each bin is calculated by dividing the cumulative count difference by the bin width:- \( [0,1] \): height = \( \frac{0.225}{1} = 0.225 \)- \( (1,3] \): height = \( \frac{0.22}{2} = 0.11 \)- \( (3,5] \): height = \( \frac{0.17}{2} = 0.085 \)- \( (5,8] \): height = \( \frac{0.12}{3} = 0.04 \)- \( (8,11] \): height = \( \frac{0.07}{3} = 0.0233 \) (approximately)- \( (11,14] \): height = \( \frac{0.105}{3} = 0.035 \)- \( (14,18] \): height = \( \frac{0.09}{4} = 0.0225 \)

Key Concepts

Empirical Distribution Function

The empirical distribution function (EDF) is a crucial concept in statistics that provides insight into how data is distributed. It is a step function that steadily increases, representing the fraction or proportion of observations less than or equal to a certain value, often denoted as \( F_n(t) \). This function is calculated from a sample dataset and plotted alongside the data points.

The EDF is particularly useful in visualizing cumulative frequencies across intervals, enhancing understanding of the data's distribution. In terms of interpreting data, the EDF helps us recognize data concentration, spots where the data stretches out, or any gaps. It's like a roadmap to trace how cumulative data "builds up" across different ranges.

For instance, imagine you have test scores ranging from 0 to 100. The EDF can tell you what percentage of students scored below various thresholds like 50, 60, or 70. In a graph, every data point has a step, showing the proportion of data up to that point.

Cumulative Frequency

Cumulative frequency refers to the sum of the frequencies for all classes up to a certain class boundary in a dataset. When you have a set of data points distributed into several bins or intervals, cumulative frequency provides the aggregate count of data points up to the end of each bin.

It's a running total that helps answer questions like "How many data points fall below a certain value?" By knowing the cumulative frequencies, we can also deduce the cumulative count differences, which highlight how much each bin contributes to the overall data.

• Suppose you have data points of students' heights and you want to find how many are shorter than a certain height.
• This is where cumulative frequency comes into play, allowing such cumulative summaries.

In histograms, this concept reveals how data is distributed across various intervals by using the cumulative totals of frequencies up to each boundary.

Bin Width

The bin width in a histogram is the measure of the difference between the upper and lower boundaries of a bin. It dictates how broad each bin interval appears on a graph, significantly affecting the histogram's shape. A crucial decision in constructing histograms, the bin width influences both clarity and detail level of the data representation.

In practice, all bins in a histogram should ideally cover equal or similar ranges unless specific attributes of the data warrant otherwise. For example, larger bin widths might mask finer details of data distribution, showing a more general overview. Conversely, narrower bin widths can exaggerate noise by showing too much detail.

• Consider bin width like the zoom level on a camera lens, offering either a wide-angle overview or a detailed close-up.
• The right bin width balances detail with clarity, ensuring the histogram effectively communicates the dataset's distribution characteristics.

Frequency Density

Frequency density in a histogram represents the height of the bars when the bin widths vary. It is calculated by dividing the frequency of the bin (or cumulative count difference) by the bin width. This measure balances the varying widths by adjusting the height of the bars accordingly, so the area of each bar correctly represents the frequency.

In cases where each bin might represent different ranges of data, frequency density becomes vital. It ensures that comparisons between the bins are consistent even when widths vary. Therefore, frequency density allows a fair comparison and accurate visual representation of data dispersal across different bins.

• Imagine bins in a histogram like plots of land; frequency density is akin to measuring how "crowded" each plot is by taking into account both its width and occupancy (frequency).
• This measure makes frequency density an essential tool for constructing reliable histograms, helping to maintain accuracy regardless of bin width variances.