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Chapter 2: Problem 14

Facebook data indicate that \(50 \%\) of Facebook users have 100 or more friends, and that the average friend count of users is \(190 .\) What do these findings suggest about the shape of the distribution of number of friends of Facebook users? \(^{17}\)

Short Answer

Expert verified
The distribution is right-skewed, with more users having fewer friends.

Step by step solution

01

Understanding the Average

The problem states that the average number of Facebook friends is 190. An average (or mean) is the sum of all values divided by the number of values. This value gives us a central tendency measure of the data, but it doesn't tell us much about the distribution's shape on its own.
02

Analyzing the Median

It is given that 50% of users have 100 or more friends, meaning the median is 100. The median is the middle value when data is arranged in order, which suggests that the lower half of the users have fewer than 100 friends, and the upper half have 100 or more.
03

Comparing Mean and Median

In this problem, the median is 100 while the mean is 190. When the mean is significantly higher than the median, it indicates a right-skewed distribution. This is because higher values (in this case, those with a very high number of friends) are pulling the mean upwards.
04

Conclusion on Distribution Shape

Since the mean is greater than the median, the distribution of number of friends is likely right-skewed. This means there are more users with fewer friends, but a smaller number with very high friend counts, which increases the average.

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Key Concepts

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

Mean vs Median
In the study of data, the mean and median are two essential measures of central tendency. Understanding these can offer insights into data characteristics and implications.
  • Mean: The mean is the arithmetic average, calculated by adding all values and dividing by the number of data points. Here, the mean number of Facebook friends is given as 190.

  • Median: The median represents the middle value when data is ordered from smallest to largest. In our case, it is 100, indicating that half of Facebook users have fewer than, and half have more than 100 friends.
By examining both the mean and the median, we can infer more about data distribution. Specifically, if the mean is notably higher than the median, it suggests that extreme high values could be pulling the average up, highlighting potential skewness in the data.
Right-Skewed Distribution
A right-skewed distribution is characterized by a tail that extends toward higher data values. This occurs when there are outliers or a few significantly large values that increase the mean compared to the median. In the case of Facebook friends, even though the median is 100, the mean is 190, demonstrating right-skewness because:
  • There are more users with fewer friends.
  • Fewer users with extremely high friend counts are enough to raise the overall average.
  • Data points that fall far above most others strongly affect the mean.
In summary, noticing a higher mean compared to the median is a good indicator that you have a right-skewed distribution in the dataset.
Central Tendency
Central tendency is a statistical measure to determine a typical value of a data set. The mean and median are part of central tendency measures, helping to summarize large datasets into meaningful numerical insights.

The choice between mean and median depends on your data distribution.
  • For symmetric distributions, the mean and median would be similar.
  • For skewed distributions, mean and median can differ noticeably, affecting how central tendency reflects real-world scenarios.
Understanding central tendency allows for better analysis and interpretation of data, especially in identifying anomalies such as right-skewed data, like seen in the Facebook friends count. Recognizing the variability between mean and median helps in establishing the overall behavioral pattern of the data set.

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