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Chapter 2: Problem 64

The mean and median for the data are the same. 3; 4; 5; 5; 6; 6; 6; 6; 7; 7; 7; 7; 7; 7; 7 Is the data perfectly symmetrical? Why or why not?

Short Answer

Expert verified
The data is not perfectly symmetrical because it is heavily skewed to the right.

Step by step solution

01

Identify Mean and Median

First, calculate the mean by adding all the numbers and then dividing by the number of data points: \( (3 + 4 + 5 + 5 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 7 + 7 + 7) / 15 = 6 \). The median is the middle value of the ordered list, which is the 8th value: 6. Thus, both mean and median equal 6.
02

Check For Symmetry

In a perfectly symmetrical distribution, each value should balance out around a central point. Count the values below and above the mean and median (both 6): Values below 6: {3, 4, 5, 5} (4 values). Values above 6: {7, 7, 7, 7, 7, 7, 7} (7 values). They are not balanced, indicating asymmetry.
03

Examine Distribution Shape

A symmetrical distribution would have an equal number of data points regularly distributed on either side of the central value. This dataset is skewed to the right, as there are more data points greater than 6 compared to those below it, confirming it is not symmetrical.

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Key Concepts

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

Mean and Median
When you calculate the mean, you are essentially finding the "average" of your dataset. For our dataset, add up all the numbers and then divide by the total count:
  • The total sum is 90.
  • Divide 90 by 15 (total number of data points), resulting in a mean of 6.
The median, on the other hand, is the middle value of the dataset when ordered sequentially. If the dataset is even, you'd average the two middle numbers together. Here, since our dataset is odd with 15 numbers, the median is the 8th number:
  • Since the list in order is 3, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, the 8th number is 6.
For this dataset, the mean and median both being 6 shows a central tendency. However, both being the same doesn't imply symmetry, as it's merely an indicator of data center.
Symmetrical Distribution
A symmetrical distribution has data points that are evenly spread around a central value. This means the shape of the data, if graphed, would be the same on both sides of the center.
  • A perfect example is the bell curve, where the left side mirrors the right side.
In a symmetrical distribution, mean, median, and mode (most frequent value) are all equal, but having these equal doesn't automatically indicate symmetry in data distribution.
In our dataset, even though the mean and median are the same, the data points do not balance evenly around 6, revealing asymmetry in distribution.
Data Analysis
Data analysis involves sorting, organizing, and interpreting datasets to uncover patterns or insights.
The first step in any data analysis is to organize the data, followed by calculating measures of center, such as mean and median.
  • Identifying skewness and symmetry also forms a crucial aspect of understanding data shape and spread.
In our example, arranging the numbers and comparing the quantity of values below and above the median provided direct insight into the distribution.
Understanding how many data points are on each side of a key value is fundamental in determining whether the dataset shows a balanced pattern or skewed tendency.
Skewness
Skewness literally means how "skewed" or tilted a dataset is from symmetry.
  • A symmetrical distribution has no skewness, while datasets leaning more on one side than the other show skewness.
  • Positive skewness, where data tails to the right, means more high values than low ones.
  • Conversely, negative skewness tails to the left.
In our dataset, observe that there are four values below 6 and seven values above it: {7, 7, 7, 7, 7, 7, 7}. This leaning suggests "right skewness"鈥攎ore data points exist to the right of our mean/median.
Recognizing skewness type aids in predicting trends and behavior, crucial for statistical modeling and interpreting data in real-world contexts.

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Use the following information to answer the next three exercises: State whether the data are symmetrical, skewed to the left, or skewed to the right. 87; 87; 87; 87; 87; 88; 89; 89; 90; 91

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