Choosing the Right Statistical Measure

Solution & Explanation

1. Choosing Between Mean and Median:

Mean:

• Definition: The mean is the arithmetic average of a dataset, calculated by summing all values and dividing by the number of values.
• Use When:
  • The data is symmetrically distributed without outliers.
  • The dataset is normally distributed.
  • You want a measure that takes into account all data points.
• Limitations:
  • Sensitive to extreme values (outliers) which can skew the mean significantly.
  • May not accurately represent the central tendency in skewed distributions.

Median:

• Definition: The median is the middle value of a dataset when it is ordered from least to greatest.
• Use When:
  • The data is skewed (either positively or negatively).
  • There are significant outliers present.
  • You require a robust measure of central tendency that is less affected by extreme values.
• Advantages:
  • Provides a better central tendency measure for skewed distributions or when outliers are present.

2. Determining Confidence Intervals:

Confidence Interval for the Mean:

• Formula:
  • For large samples (n > 30):
    • $CI = \bar{x} \pm z_{\alpha/2} \left( \frac{\sigma}{\sqrt{n}} \right)$
  • For small samples (n < 30):
    • $CI = \bar{x} \pm t_{\alpha/2, n-1} \left( \frac{s}{\sqrt{n}} \right)$
• Components:
  • $\bar{x}$: Sample mean
  • $\sigma$: Population standard deviation (or $s$, sample standard deviation if $\sigma$ is unknown)
  • $n$: Sample size
  • $z_{\alpha/2}$: z-score for desired confidence level (e.g., 1.96 for 95%)
  • $t_{\alpha/2, n-1}$: t-score for desired confidence level with $n-1$ degrees of freedom

Confidence Interval for the Median:

• Method:
  • Since the median does not follow a normal distribution, bootstrapping is often used.
  • Bootstrapping Approach:
    1. Resample the dataset with replacement multiple times (e.g., 1000 times).
    2. Calculate the median for each resampled dataset.
    3. Determine the standard error (SE) of these medians.
    4. Construct the confidence interval: $\text{Median} \pm 1.96 \times \text{SE}$.

Conclusion:

• Mean is suitable for symmetric, outlier-free datasets, while median is advantageous for skewed distributions with outliers.
• Confidence intervals provide a range within which the true population parameter is likely to fall, with specific methods for calculating them based on whether the mean or median is used.