Question

Symmetric or Skewed? Based on the values of the mean and the median, decide whether the data set is skewed right, skewed left, or approximately symmetric. $$ \bar{x}=5.38 ; m=5.34 $$

Short answer

Mean value: 5.38 Median value: 5.34

Step-by-step solution

Recap the relationships between the mean and median in skewed and symmetric distributions

When analyzing a data set, we can make some conclusions about the shape of its distribution based on the relationship between the mean and the median: - If the mean is greater than the median, the data is skewed right. - If the mean is less than the median, the data is skewed left. - If the mean is approximately equal to the median, the data is approximately symmetric.

Compare the mean and the median

Now, let's compare the given mean and median values: Mean (\(\bar{x}\)) = 5.38 Median (\(m\)) = 5.34 As we can see, the mean and the median are quite close.

Make the conclusion

Since the mean (5.38) is very close to the median (5.34), we can conclude that the data set is approximately symmetric.

Key concepts

Mean

The mean of a data set, often represented as \( \bar{x} \), is the average of all the values in the set. It is calculated by adding up all the individual values and then dividing by the number of values. For the purpose of analyzing distributions, the mean serves as a measure of central tendency, giving us insight into the 'center' of the data.

• If all values in a data set are equal, the mean will be that value.
• The mean is sensitive to outliers, meaning extreme values can affect it significantly.

Understanding the mean is crucial because it helps us make decisions about the distribution's symmetry. Comparing it to the median can reveal the skewness of the data. In our example, with a mean of 5.38, any significant deviation from the median could suggest skewness.

Median

The median is a simple yet powerful statistic. It is the value that separates the higher half from the lower half of a data set. If you arrange the data points in order, the median is right in the middle.

• In datasets with an odd number of observations, the median is the middle value.
• In datasets with an even number of observations, it is the average of the two middle values.

The median is particularly handy because it is resistant to outliers. Unlike the mean, which can be skewed by extreme values, the median tends to provide a better measure of center for skewed distributions. In our situation, a median of 5.34 closely matching the mean suggests that skewness is minimal.

Skewness

Skewness tells us about the asymmetry of the data distribution. It indicates whether the data leans more toward the left or right of the mean.

• Right-skewed (positive skew) means the mean is greater than the median. This suggests a longer tail on the right.
• Left-skewed (negative skew) means the mean is less than the median. This indicates the data has a longer tail to the left.
• When the mean and median are close, it's a sign of a symmetric distribution.

Our analysis showed the mean is 5.38 and the median is 5.34, a slight difference suggesting the distribution is approximately symmetric, showing minimal skewness.

Symmetric Distribution

A symmetric distribution is balanced, with both sides mirroring each other. In such distributions, the mean and median are approximately equal, indicating no skewness.

• Symmetric distributions often resemble a bell curve, known as a normal distribution.
• Both mean and median give similar information about the central tendency.

Different real-world data sets can exhibit symmetry. Checking close proximity between mean and median is crucial to determining symmetry. In our example, a mean of 5.38 and a median of 5.34 suggest the dataset is nearly symmetric, indicating that the distribution's tails are balanced.