2.1 Measures of central tendency and dispersion

Understanding data is crucial in decision-making. Central tendency measures like mean, median, and mode help summarize datasets. These tools provide quick insights into typical values, allowing managers to grasp the core of their data.

Dispersion measures like range, variance, and standard deviation reveal data spread. By considering both central tendency and dispersion, decision-makers can better understand data variability and make more informed choices in various business scenarios.

Measures of Central Tendency

Measures of central tendency

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• Mean calculates arithmetic average of dataset values $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$ represents balance point (average test score in a class)
• Median finds middle value when data arranged in order less affected by outliers (housing prices in a neighborhood)
• Mode identifies most frequently occurring value can have multiple modes (shoe sizes in a store)

Measures of data dispersion

• Range measures difference between maximum and minimum values $Range = Max(x) - Min(x)$ provides quick spread measure (temperature range in a day)
• Variance calculates average of squared deviations from mean $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$ quantifies variability (stock price fluctuations)
• Standard deviation computes square root of variance $s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}$ measures average distance from mean (customer satisfaction ratings)

Selection of central tendency measures

• Consider data type (nominal, ordinal, interval, ratio) and distribution shape (symmetric, skewed, uniform)
• Mean works best for interval or ratio data with symmetric distribution (height measurements)
• Median suits ordinal, interval, or ratio data preferred for skewed distributions (income levels)
• Mode applies to nominal data useful for categorical or discrete numerical data (favorite colors)

Impact of outliers on statistics

• Mean highly sensitive to outliers pulled towards extreme values (average income skewed by billionaires)
• Median resistant to outliers remains stable with extreme values (house prices in mixed neighborhood)
• Mode generally unaffected by outliers (most common car color)
• Range highly sensitive to outliers increased by extreme values (age range in a classroom with one elderly student)
• Variance and standard deviation increased by outliers due to squared differences (test scores with one perfect score)
• Strategies for handling outliers:
  1. Identify potential data entry errors
  2. Use robust measures (median, interquartile range)
  3. Analyze data with and without outliers