Averages: Mean, Median, and Mode

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1. Mean

• Definition: The sum of all values divided by the number of values.
• Formula:
  [
  {Mean} = \frac{{Sum of all values}} / {{Number of values}}
  ]
• Example:
• Prices of loaves of bread: $1, $1.20, $1.10.
• Total: (1 + 1.20 + 1.10 = 3.30).
• Mean = (3.30 \div 3 = $1.10).

Key Uses: Finding an average value for a dataset, predicting trends, or comparing groups.

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2. Median

• Definition: The middle value in a ranked (sorted) dataset.
• Steps:
• Arrange values in ascending order.

• Identify the middle value:

  • For an odd number of values: Single middle value.
  • For an even number of values: Average of the two middle values.

• Example 1 (Odd dataset):
  Dataset: 6, 13, 67, 45, 2.
  Ranked: 2, 6, 13, 45, 67.
  Median = 13.

• Example 2 (Even dataset):
  Dataset: 6, 13, 67, 45, 2, 7.
  Ranked: 2, 6, 7, 13, 45, 67.
  Median = ((7 + 13) \div 2 = 10).

Key Uses: Handling skewed data to represent the "middle" value.

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3. Mode

• Definition: The most frequently occurring value(s) in a dataset.
• Steps:
• Count the frequency of each value.

• Identify the value(s) with the highest frequency.

• Example 1 (Single mode):
  Balloon colors: 18 red, 12 blue, 24 orange, 25 purple, 21 green.
  Mode = Purple.

• Example 2 (No clear mode):
  Speeds: 40, 34, 42, 38, 41, 50, 48, 49, 33, 47.

• All values are unique No mode.
• Grouped Mode: Create categories:
  • 30–32: 0, 33–35: 2, 36–38: 1, 39–41: 2, 42–44: 1, 45–47: 1, 48–50: 3.
  • Mode = Category 48–50 Midpoint: 49.

Key Uses: Identifying the most common occurrences, such as customer preferences or frequently observed trends.

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Comparison of Mean, Median, and Mode

| Measure | Best for | Limitations |
|-----------------|---------------------------------------------------|------------------------------------------------|
| Mean | Evenly distributed data, detailed comparisons. | Affected by extreme values (outliers). |
| Median | Skewed or non-symmetrical data. | May not fully represent all data points. |
| Mode | Categorical data or frequent occurrences. | Less useful with no repetition or multiple modes. |

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Practical Applications

• Business: Calculate average sales or revenue trends.
• Education: Analyze student performance using mean and median scores.
• Science: Use mode to study repeated observations in experiments.
• Daily Life: Plan budgets or predict future expenses based on averages.

By understanding these measures, you can better interpret and analyze data to make informed decisions!

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