Adding a Constant to Dataset Values

Solution & Explanation

When a constant $c$ is added to each data point in a dataset, it impacts various sample statistics differently. Let's explore the impact on each of the requested statistics:

1. Mean

• Effect: The mean will increase by the constant $c$.
• Explanation: The mean is calculated as the sum of all data points divided by the number of data points. If each data point is increased by $c$, then the total sum increases by $n \times c$ (where $n$ is the number of data points). Thus, the new mean becomes the old mean plus $c$.

2. Median

• Effect: The median will increase by the constant $c$.
• Explanation: The median is the middle value in an ordered dataset. Adding $c$ to each data point shifts the entire dataset by $c$, so the position of the median remains unchanged, but its value increases by $c$.

3. Mode

• Effect: The mode will increase by the constant $c$.
• Explanation: The mode is the most frequently occurring value in the dataset. Adding $c$ to each data point shifts all values equally, including the mode, so the mode value increases by $c$ but its frequency remains the same.

4. Range

• Effect: The range remains unchanged.
• Explanation: The range is calculated as the difference between the maximum and minimum values in the dataset. Since both the maximum and minimum values are increased by $c$, the difference between them remains the same.

5. Variance

• Effect: The variance remains unchanged.
• Explanation: Variance measures the spread of the data points around the mean. Since adding $c$ to each data point shifts the entire dataset without altering the relative distances between data points, the spread remains the same. Thus, the variance is unaffected.

Summary

• Central Tendency Measures (mean, median, mode): These will increase by the constant $c$ because they are measures of the center of the dataset, which shifts when $c$ is added.
• Spread Measures (range, variance): These remain unchanged because they measure the dispersion of the dataset, which is unaffected by a uniform shift.

Understanding how different statistics react to transformations such as adding a constant is crucial for data preprocessing and analysis, ensuring accurate interpretations and decisions based on the data.