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The following data give the weights (in pounds) of a random sample of 44 college students. (Here \(\mathrm{F}\) and \(\mathrm{M}\) indicate female and male, respectively.) \(\begin{array}{llllll}123 \mathrm{~F} & 195 \mathrm{M} & 138 \mathrm{M} & 115 \mathrm{~F} & 179 \mathrm{M} & 119 \mathrm{~F} \\ 148 \mathrm{~F} & 147 \mathrm{~F} & 180 \mathrm{M} & 146 \mathrm{~F} & 179 \mathrm{M} & 189 \mathrm{M} \\ 175 \mathrm{M} & 108 \mathrm{~F} & 193 \mathrm{M} & 114 \mathrm{~F} & 179 \mathrm{M} & 147 \mathrm{M} \\ 108 \mathrm{~F} & 128 \mathrm{~F} & 164 \mathrm{~F} & 174 \mathrm{M} & 128 \mathrm{~F} & 159 \mathrm{M} \\ 193 \mathrm{M} & 204 \mathrm{M} & 125 \mathrm{~F} & 133 \mathrm{~F} & 115 \mathrm{~F} & 168 \mathrm{M} \\ 123 \mathrm{~F} & 183 \mathrm{M} & 116 \mathrm{~F} & 182 \mathrm{M} & 174 \mathrm{M} & 102 \mathrm{~F} \\ 123 \mathrm{~F} & 99 \mathrm{~F} & 161 \mathrm{M} & 162 \mathrm{M} & 155 \mathrm{~F} & 202 \mathrm{M} \\ 110 \mathrm{~F} & 132 \mathrm{M} & & & & \end{array}\) Compute the mean, median, and standard deviation for the weights of all students, of men only, and of women only. Of the mean and median, which is the more informative measure of central tendency? Write a brief note comparing the three measures for all students, men only, and women only.

Step by step solution

01

Organize the data

First organize the weight data for men and women separately, then combine them for all students data. Sorting them in ascending order will be helpful to calculate median later.

02

Calculate mean

The formula to find the mean or average is sum of the data divided by the number of data points. Do the math separately for Male students, Female students, and All students.

03

Calculate median

To find the median, if the data count is odd, the median is the middle number. if it's even, then it's the average of the two middle numbers. Recall that you already sorted the data in ascending order in step 1. So now it'll be easy to find the median.

04

Calculate standard deviation

To find the standard deviation, first find out the variance: subtract the mean from each data point and square the result, then find their mean, and finally, the standard deviation is the square root of the variance. Do this for each category.

05

Analyze results

Now look at the results for mean, median and standard deviation of each group. Which measure of central tendency, mean or median, provides the most information will depend on the nature of the distribution. If the distribution is symmetrical, the mean and median should be very close. If they're not close, then the data might be skewed and the median might provide more useful information. Compare these measures for all students, men only, and women only.

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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 a fundamental measure of central tendency. It's used to get a general sense of the 'central' value for a set of numbers. To calculate the mean, add up all the values and divide by the number of values.
For our weight data, you'll calculate the mean separately for all students, men, and women.

• For All Students: Sum up all weights, then divide by 44.
• For Men Only: Sum the weights, then divide by the total number of men in the sample.
• For Women Only: Do the same as above but only with the female students' weights.

Calculating the mean gives you an overall "balance point" for the data, showing the general trend.

Median Calculation

The median is another central measure that often gives a better sense of the middle value when data sets are skewed. The process to find the median is straightforward:
Arrange the data in ascending order, then identify the middle point. If the dataset count is odd, pick the central number. If even, take the average of the two central numbers.

• For all students, simply put the weights in order and locate the middle.
• Repeat the same for male and female groups independently.

Unlike the mean, the median is less affected by extreme values (also known as outliers), making it a reliable indicator when data have anomalies.

Standard Deviation

The standard deviation is a key metric that describes the spread or variability of data around the mean. It tells you how much variation exists from the average.
Here's how to calculate:

• Find the mean of the data set.
• Subtract the mean from each data point and square the result.
• Calculate the mean of these squared differences.
• Take the square root of that mean to get the standard deviation.

A smaller standard deviation signifies that data points tend to be closer to the mean. A larger deviation suggests more spread out data, indicating variability.

Data Analysis

Performing data analysis involves examining results from the calculations above and integrating insights. This helps in understanding the data better.
When you compare the mean and median, note which one is closer to most of your data. The median is beneficial when there's a large difference between it and the mean, as it indicates skewness in data distribution.
Consider standard deviation for a sense of data spread. In our weight example, seeing whether men or women have a higher deviation can shed light on weight distribution.
This analytical process shows how each measure offers different insights into data tendencies, balance, and variability.