Question

In a sample of hospital patients, the mean age is found to be significantly lower than the median. Which of the following best describes this distribution? (A) Skewed right (B) Skewed left (C) Normal (D) Bimodal

Short answer

B) Skewed left

Step-by-step solution

Understand Mean and Median

The mean is the average of a set of numbers, while the median is the middle value when the numbers are arranged in ascending order.

Compare Mean and Median

If the mean is significantly lower than the median, this indicates that the data has more high values that skew the mean towards the lower end.

Identify Distribution Type

In a distribution where the mean is lower than the median, it is characterized as being skewed to the left. This means that the tail of the distribution is on the left side.

Select the Correct Answer

Based on the analysis, the best description of the distribution is (B) Skewed left.

Key concepts

skewness

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It indicates whether the data points are spread out more to one side of the mean than the other. There are two types of skewness:
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**Positive Skewness**: When the tail on the right side (higher values) of the distribution is longer or fatter, the distribution is said to be positively skewed or right-skewed. This usually happens when extreme high values pull the mean upwards.

**Negative Skewness**: When the tail on the left side (lower values) of the distribution is longer or fatter, the distribution is negatively skewed or left-skewed. In this case, extreme low values pull the mean downwards.

Understanding skewness helps in identifying the direction of outliers and the overall shape of the distribution.

mean and median comparison

The mean and median are both measures of central tendency but they capture different aspects of a data set. The **mean** is the arithmetic average of all numbers in the data set, calculated by summing all the values and dividing by the total number of values. The **median**, on the other hand, is the middle value when the data points are arranged in ascending order.

When comparing these two:

• **If the mean is equal to the median**: The distribution is symmetric. Examples include a normal distribution where data points are evenly distributed around the center.
• **If the mean is greater than the median**: The distribution is right-skewed (positively skewed). This means that there are outliers or a longer tail on the right side.
• **If the mean is less than the median**: The distribution is left-skewed (negatively skewed). This indicates outliers or a longer tail on the left side.

Understanding the relationship between mean and median provides insight into the underlying distribution of the data.

left-skewed distribution

A left-skewed distribution, also known as a negatively skewed distribution, is characterized by a longer or fatter tail on the left side. In simpler terms, there are a few extremely low values pulling the mean to the left of the median.
Key features of a left-skewed distribution include:

• **Mean less than median**: As seen in the original problem where the mean age of hospital patients is significantly lower than the median. This discrepancy is due to the presence of lower extremes.
• **Long left tail**: Most of the data points are concentrated on the right, but there are still values that extend to the left side.
• **Asymmetry**: The peak of the distribution is shifted to the right, making the distribution asymmetrical.

Recognizing a left-skewed distribution is essential for accurate data interpretation and analysis.

data analysis

Data analysis involves systematically applying statistical and logical techniques to describe and illustrate, condense and recap, and evaluate data. Analyzing the distribution type is crucial because it influences how we interpret and act on data insights.

Steps to perform effective data analysis:

• **Collect Data**: Gather the necessary data from various sources.
• **Clean Data**: Remove or correct any inaccuracies or inconsistencies in the data.
• **Analyze Distributions**: Determine the shape of the data distribution (e.g., skewness, symmetry).
• **Identify Patterns**: Look for trends, patterns, and outliers in the data.
• **Compare Metrics**: Compare key statistical measures like mean and median to gain further insights.

The insights gained from data analysis can be used for making informed decisions, forecasting, and creating strategies. Understanding the underlying distribution helps in determining the appropriate statistical methods to use.