91Ó°ÊÓ

Are the data Normal? Monsoon rains. Here are the amounts of summer monsoon rainfall (millimeters) for India in the 100 years 1901 to \(2000:^{16}\) :II MONSOON $$ \begin{array}{llllllrrrr} 722.4 & 792.2 & 861.3 & 750.6 & 716.8 & 885.5 & 777.9 & 897.5 & 889.6 & 935.4 \\\ 736.8 & 806.4 & 784.8 & 898.5 & 781.0 & 951.1 & 1004.7 & 651.2 & 885.0 & 719.4 \\\ 866.2 & 869.4 & 823.5 & 863.0 & 804.0 & 903.1 & 853.5 & 768.2 & 821.5 & 804.9 \\\ 877.6 & 803.8 & 976.2 & 913.8 & 843.9 & 908.7 & 842.4 & 908.6 & 789.9 & 853.6 \\\ 728.7 & 958.1 & 868.6 & 920.8 & 911.3 & 904.0 & 945.9 & 874.3 & 904.2 & 877.3 \end{array} $$ \(\begin{array}{llllllll}739.2 & 793.3 & 923.4 & 885.8 & 930.5 & 983.6 & 789.0 & 889.6\end{array}\) \(\begin{array}{lllllll}1020.5 & 810.0 & 858.1 & 922.8 & 709.6 & 740.2 & 860.3\end{array}\) \(\begin{array}{lllllll}887.0 & 653.1913 .6 & 748.3 & 963.0 & 857.0 & 883.4909 .5 & 708.0\end{array}\) \(\begin{array}{lllllll}852.4735 .6955 .9836 .9 & 760.0 & 743.2 & 697.4961 .7 & 866.9908 .8\end{array}\) \(\begin{array}{lllllll}784.7 & 785.0 & 896.6 & 938.4 & 826.4 & 857.3 & 870.5\end{array}\) (a) Make a histogram of these rainfall amounts. Find the mean and the median. (b) Although the distribution is reasonably Normal, your work shows some departure from Normality. In what way are the data not Normal?

Step by step solution

01

Organize the Data

First, list all the rainfall data to get an overview. Ensure to remove any potential formatting errors during the extraction process.

02

Plot a Histogram

Use the data to construct a histogram. This will be done by dividing the range of rainfall amounts into bins and counting how many data points fall into each bin. This helps in visually assessing the shape of the distribution.

03

Calculate the Mean

Find the mean by adding all the rainfall amounts together and dividing by the total number of observations. If the total of the amounts is represented as \( \Sigma X \) and the number of observations is \( N \), then Mean \( \mu = \frac{\Sigma X}{N} \).

04

Calculate the Median

To find the median, first sort the rainfall amounts in ascending order. The median is the middle value of this ordered list if the number of data points is odd, or the average of the two middle values if even.

05

Analyze the Normality of the Data

Compare the histogram's shape to a Normal distribution (bell curve). Check for symmetry, and the mean should ideally be equal to the median. Alternatively, use statistical tests like the Shapiro-Wilk test for a more quantitative analysis. Determine where the data deviates from normality: skewness, kurtosis, or outliers.

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Key Concepts

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

Histogram Construction

One effective way to understand data distribution is by constructing a histogram. This graphical representation allows you to visualize the frequencies of different ranges of data, also known as bins. In the context of the monsoon rainfall data, you start by organizing the rainfall measurements from the smallest to the largest value. Next, you'll need to determine the number and width of the bins. Typically, the number of bins can be decided based on the audience's preference or using formulas like Sturges' rule, which calculates the optimal number based on the number of observations. Once the bins are determined, plot the histogram:

• Each bar represents the frequency of data points within the bin.
• The height of the bar illustrates how many data points fall within each range of rainfall amounts.
• Ensure that all data points are accounted for within the range of your histogram.

This histogram will help you see if there's a bell-shaped, uniform, or skewed distribution. A normal distribution would typically form a bell curve.

Mean and Median Calculation

Calculating the mean and median of a dataset gives insight into its central tendency. Let's start with the mean, which is the average value of the data. Add up all the rainfall measurements and divide by the number of years to find the mean. This formula is expressed as \( \mu = \frac{\Sigma X}{N} \), where \( \Sigma X \) is the total millimeters of rainfall, and \( N \) is the 100 years over which measurements were taken. Next, calculate the median to find the middle point in an ordered list of the data. For a sorted list:

• If the number of observations \( N \) is odd, the median is the middle value.
• If \( N \) is even, the median is the average of the two central numbers.

While the mean provides a picture of the dataset's overall trend, the median can be more informative for skewed data, offering a better sense of the center.

Normal Distribution Analysis

Detecting normality in a dataset involves comparing its distribution to the ideal bell curve shape. In the case of the rainfall data, begin by analyzing the histogram. A normal distribution is symmetric around the mean, with the mean, median, and mode all roughly equal. However, real-world data rarely fits perfectly into this mold. Look for:

• Symmetry: A normal distribution will have a histogram that's mirrored on either side of the mean.
• Tails: Check if the tails of the data (extreme high and low points) taper off smoothly.
• Peaks: The peak should occur near the mean value.

Additionally, you can use statistical tests like the Shapiro-Wilk test to provide a quantitative measure of normality. Outliers, skewness (when one tail is longer), and kurtosis (tailedness or peakedness) can indicate departures from normality. These characteristics should prompt closer investigation as they could affect further statistical analyses and outcomes.