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Difficulty: EasyMeasures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

A dataset consists of 15 numerical values with a mean of 50 and a standard deviation of 8. If a constant value of 5 is added to every number in the dataset, what is the standard deviation of the resulting dataset?

  1. 8Answer
  2. B
    13
  3. C
    40
  4. D
    55
  5. E
    64

Answer

The standard deviation of the resulting dataset is 8.
Adding a constant to every data point in a set shifts the central location (the mean and median) by that constant, but leaves the distances between points unchanged. Because standard deviation depends solely on the distances between each point and the mean, the standard deviation remains 8.

Step-by-Step Solution

1
Recall the effect of adding a constant to data points on measures of dispersion.
Measures of dispersion (range, IQR, standard deviation) describe the spread or scatter of data relative to its center.
When a constant kk is added to every value xix_i, the new mean becomes mean+k\text{mean} + k.
2
Evaluate the distance of each transformed data point from the new mean.
(xi+k)−(mean+k)=xi−mean(x_i + k) - (\text{mean} + k) = x_i - \text{mean}.
The deviation of each point from the mean remains identical to its original deviation.
3
Determine the standard deviation of the new dataset.
The standard deviation is unchanged and remains 8.
Because all individual deviations from the mean are preserved, the average distance from the mean does not change.

Key Concept

Invariance of standard deviation under addition of a constant
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Topics

Measures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

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