91Ó°ÊÓ

Lengths of bus routes for any particular transit system will typically vary from one route to another. The article "Planning of City Bus Routes" (J. of the Institution of Engineers, 1995: 211-215) gives the following information on lengths \((\mathrm{km})\) for one particular system: $$ \begin{array}{lccccc} \text { Length } & 6-<8 & 8-<10 & 10-<12 & 12-<14 & 14-16 \\ \text { Frequency } & 6 & 23 & 30 & 35 & 32 \\ \text { Length } & 16-<18 & 18-<20 & 20-<22 & 22-<24 & 24<26 \\ \text { Frequency } & 48 & 42 & 40 & 28 & 27 \\ \text { Length } & 26-<28 & 28-<30 & 30-<35 & 35-<40 & 40-<5 \\ \text { Frequency } & 26 & 14 & 27 & 11 & 2 \end{array} $$ a. Draw a histogram corresponding to these frequencies. b. What proportion of these route lengths are less than 20 ? What proportion of these routes have lengths of at least 30 ? c. Roughly what is the value of the 90 th percentile of the route length distribution? d. Roughly what is the median route length?

Step by step solution

01

Understanding the Data

The dataset represents bus route lengths in kilometers and their respective frequencies in different length intervals. The aim is to perform tasks like plotting a histogram, calculating proportions, and finding percentile and median values.

02

Plotting the Histogram

Using the given data, plot a histogram where each bar represents a length interval, and the height of each bar represents the corresponding frequency. The x-axis will contain the route length intervals, and the y-axis will show the frequency of each interval.

03

Finding the Proportion for Part (b)

1. To find the proportion of routes less than 20 km, sum the frequencies for the intervals 6-8, 8-10, 10-12, 12-14, 14-16, 16-18, and 18-20. 2. To find the proportion of routes at least 30 km, sum the frequencies for the intervals 30-35, 35-40, and 40-45. 3. Divide each total by the sum of all frequencies to get the respective proportions.

04

Calculating the 90th Percentile

First, determine the cumulative frequency distribution. The 90th percentile position is given by 0.9 times the sum of all frequencies. Identify the interval corresponding to this position and approximate the 90th percentile length based on cumulative frequencies.

05

Finding the Median

The median corresponds to the 50th percentile position, which is 0.5 times the total frequency. Using the cumulative frequency distribution, locate the interval where the median lies and approximate the route length that corresponds to this cumulative frequency position.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Histogram

A histogram is a type of bar graph that represents the frequency distribution of a dataset. In our exercise, each bar in the histogram corresponds to a range of bus route lengths, such as 6-8 km or 8-10 km.
The height of each bar shows how many routes fall within that specific length interval.
To plot a histogram:

• Place the route length intervals on the x-axis.
• On the y-axis, indicate the frequency of each interval.
• Draw bars for each interval where the height is proportional to the frequency.

This visual representation helps in quickly identifying which length ranges are most common, making it easier to analyze the distribution of bus route lengths.

Percentiles

Percentiles are values below which a certain percentage of data falls and are key in understanding the spread of a dataset. For instance, the 90th percentile shows us the length below which 90% of the bus routes lie.
To find a percentile:

• First, calculate the cumulative frequency for the dataset.
• Identify the position of the desired percentile by multiplying the total number of routes by the percentile's decimal value (e.g., 0.9 for the 90th percentile).
• Find the length interval where this cumulative frequency position falls.
• Estimate the length by determining within the interval how far into the percentile we are.

Percentiles help in assessing the data spread, identifying outliers, and making decisions based on distribution patterns.

Median

The median is a measure of central tendency that represents the middle value in a dataset when the values are arranged in order. For bus route lengths, the median tells us the length at which half of the routes are shorter, and half are longer.
To find the median:

• Calculate the total number of observations.
• Determine the median position, which is 0.5 times the total frequency.
• Use the cumulative frequency to locate which interval contains the median route length.
• Approximate the length within this interval where the 50% position lies.

The median provides a clear picture of the typical length of a bus route, less influenced by outliers compared to the mean.

Frequency Distribution

Frequency distribution is a tabular summary showing the number of observations (frequencies) within specific intervals (bins). In the context of our exercise, it tabulates how the bus routes are distributed across various length intervals (e.g., 6-8 km, 8-10 km, etc.).
To interpret frequency distribution:

• Identify the intervals with the highest frequencies to understand where most data points lie.
• Consider the shape of the distribution to see if it is skewed or symmetrical.
• Analyze the breadth (or spread) of intervals to understand variability in route lengths.

Frequency distributions provide essential insights into the data's overall pattern and are foundational in graphical representations like histograms.