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The Belmont Stakes is the final race in the annual Triple Crown of thoroughbred horse racing. The race is \(1.5\) miles in length, and the record for the fastest time of 2 minutes, 24 seconds is held by Secretariat, the 1973 winner. We compared Secretariat's time from 1973 with the time of each winner of the Belmont Stakes for the years \(1999-2011\). The following data represent the differences (in seconds) between each winner's time for the years \(1999-2011\) and Secretariat's time in 1973 . For example, the 1999 winner took \(3.80\) seconds longer than Secretariat to finish the race. \(\begin{array}{lllllllllllllll}3.80 & 7.20 & 2.80 & 5.71 & 4.26 & 3.50 & 4.75 & 3.81 & 4.74 & 5.65 & 3.54 & 7.57 & 6.88\end{array}\) a. Calculate the mean and median. Do these data have a mode? Why or why not? b. Compute the range, variance, and standard deviation for these data.

Step by step solution

01

Calculate the Mean

The mean is the average of all numbers and is computed as the sum of all the numbers divided by the total count of numbers. Summing all 13 race times and dividing by 13 will give the mean time.

02

Calculate the Median

The median is the value separating the higher half from the lower half of a data sample. Arrange all 13 values of difference in time in increasing order and select the one in the middle. Since there are 13 values, the 7th value will be the median.

03

Determine the Mode

The mode is the number that appears most often in a data set. A data set may have one mode, more than one mode, or no mode at all. Check the 13 values to see if any value appears more than once.

04

Compute the Range

The range of a set of data is the difference between the highest and lowest values in the set. To calculate the range, subtract the minimum time difference from the maximum time difference.

05

Calculate the Variance

The variance is a measure of how spread out the numbers in the data are. It is the average of the squared differences from the mean. To calculate the variance, first, calculate the difference from the mean for each number in the data set and then square the result. Do this for all numbers and then calculate the average of these squared differences.

06

Compute the Standard Deviation

The standard deviation is a measure of how spread out numbers in the data are. It is the square root of the variance. Calculate the square root of the variance obtained in Step 5.

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Key Concepts

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

Mean Calculation

The mean, often known as the average, is one of the most straightforward descriptive statistics tools to understand. It allows us to find a central value in our data set. Calculating the mean involves adding up all the numbers in your data set and then dividing by the total amount of numbers you have added. For our set, this involves summing up the numbers representing the differences in race times and then dividing by the total number of races, which is 13.
The formula for the mean is as follows: \[ \text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n} \]where \(x_i\) represents each value in the data set and \(n\) is the total number of values. This provides a single value around which the other numbers are distributed (although not perfectly, as data can be skewed). It helps give a quick overview of our data's performance.

Median Determination

In descriptive statistics, the median provides us with the "middle" value of a dataset, highlighting the center of the set when arranged in order. To find the median, arrange the data points in ascending order and identify the middle number. With 13 values in our example, we simply find the seventh value, as it effectively divides the dataset into two equal parts.
In our context: - Order the differences from smallest to largest. - The seventh value is your median.
This number particularly helps us understand the dataset better when there are outliers or skewed data that could distort the mean.

Mode in Data Sets

The mode of a dataset captures the most frequently occurring value, showing what number, if any, is more common than others. Sometimes there will be no mode when each number appears only once, as is the case with our dataset of race times.
- To find the mode, list out each number from the dataset.
- Determine if any numbers appear more than once. If they do, those are your modes. Otherwise, there is no mode.
A set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all. Recognizing modes can often reveal trends or significant data points that may not be evident through the mean or median alone.

Range

The range gives us a basic idea about the spread of our data by calculating the difference between the highest and lowest values in the set. It provides a quick view into how much variability there is.
To calculate the range:

• Identify the maximum and minimum numbers in your dataset.
• Subtract the minimum value from the maximum value.

In our example, this helps show how wide the differences in race performance times are. Although simple, the range can often be influenced significantly by outliers.

Variance

Variance measures how much the data values deviate from the mean, offering a sense of data's spread and consistency. A higher variance indicates that the data points are more spread out from the mean, whereas a lower variance suggests they are closer.
To compute variance:

• Firstly, subtract the mean from each number to find the deviation for each data point.
• Square each deviation to eliminate negative values.
• Then, sum all squared deviations and divide by the number of data points to get the average.

This process gives us the variance, quantifying the extent of dispersion within the dataset.

Standard Deviation

The square root of the variance gives us the standard deviation, an essential tool for understanding data spread. It is expressed in the same units as the original data, providing us a more comprehendible measure of variability.
To calculate:

• First, calculate the variance as explained in the previous section.
• Take the square root of the computed variance.

A smaller standard deviation means that most data points are close to the mean, while a larger one suggests broader spread, making it a very useful tool for assessing data consistency.