Statistics: Mean, Weighted Mean, and Median

Mean

• The mean is the average of a dataset.
• The mean is calculated by summing all values in the dataset and dividing by the number of values.
• The symbol for the population mean is '𝜇'
• If the dataset represents a sample, the symbol for the sample mean is 'x̅'

Weighted Mean

• The weighted mean is an arithmetic mean where each value is weighted according to its importance.
• The weighted mean formula is identical for population and sample data:
  • 𝜇w or Xw = ∑(wX) / ∑w
• The weighted mean is calculated by multiplying each value (X) by its corresponding weight factor (w), summing the products, and then dividing by the sum of the weights.
• The weighted mean is used to calculate final grades in a course when the number of units per subject is not equal.

Median

• The median is the central value in a data set when arranged in ascending order.
• The median divides the data into two equal parts.
• The median is useful for summarizing skewed distributions as it is not sensitive to extreme values.
• The median is equal to the mean for symmetric distributions.
• Data must be at least ordinal to calculate a median.

Median Calculation

• If the number of data points (N or n) is odd, the median is the middle value.
• If the number of data points (N or n) is even, the median is the average of the two middle values.
• The symbol for the population median is "𝜇̃".
• The symbol for the sample median is "x̃"

Relationship Between the Mean and the Median

• The shape of the distribution is crucial when choosing a representative measure of central tendency.
• The relationship between the mean and the median can help determine the shape of the distribution.
• Symmetric distribution:* mean = median
• Positively skewed distribution:* mean > median
• Negatively skewed distribution:* mean < median

Skewness

• Skewness measures the asymmetry of a distribution.
• A positive skew indicates a longer tail on the right side of the distribution.
• A negative skew indicates a longer tail on the left side of the distribution.
• If skewness is less than -1 or greater than +1, the distribution is highly skewed.
• If skewness is between -1 and -0.5, or between +0.5 and +1, the distribution is moderately skewed.
• If skewness is between -0.5 and +0.5, the distribution is approximately symmetric.

Empirical Rule

• The empirical rule applies to bell-shaped distributions.

• It helps estimate the percentage of data values within a specified number of standard deviations from the mean.

  • 68%: Approximately 68% of values lie within one standard deviation of the mean (𝜇 ± 1σ)
  • 95%: Approximately 95% of values lie within two standard deviations of the mean (𝜇 ± 2σ)
  • 99.7%: Approximately 99.7% of values lie within three standard deviations of the mean (𝜇 ± 3σ)

Bell-Shaped Curve (Normal Curve)

• The total area under the curve and above the horizontal line is 1 or 100%.
• The curve is symmetric.
• The area between similarly distanced points on the x-axis from the mean are equal.

Outlier

• An outlier is a data point that is significantly different from other data points in the dataset.
• Outliers can have a strong impact on the mean.
• Outliers can be identified by using a box plot or by calculating the interquartile range (IQR).

This quiz covers the concepts of mean, weighted mean, and median in statistics. You'll learn how to calculate each measure and understand their significance in data analysis. Perfect for those looking to reinforce their knowledge of these fundamental statistical concepts.