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Statistics and Probability - Measures of Central Tendency (Mean, Median, Mode)

Grade 7IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Measures of Central Tendency provide a single value that represents the center or 'typical' value of a dataset. Imagine a dot plot where the data points cluster around a central peak; these measures help identify exactly where that peak or balance point lies.

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The Mean is the arithmetic average, calculated by adding all values and dividing by the total count nn. Visually, it represents the 'leveling out' of the data, as if you moved values from high bars in a histogram to fill the gaps in lower bars until all bars reached the same height.

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The Median is the middle value when data is arranged in ascending or descending order. If you visualize the data points as a line of people ordered by height, the median is the height of the person standing exactly in the middle of the line, dividing the group into two equal halves.

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The Mode is the value that occurs most frequently in a data set. In a bar graph or frequency chart, the mode corresponds to the tallest bar, indicating the most popular or common category. A data set can have one mode, multiple modes (bimodal/multimodal), or no mode at all.

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The Range measures the spread of the data by subtracting the smallest value from the largest value. On a number line, the range is the physical distance between the data point furthest to the left and the point furthest to the right.

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Outliers are data points that are significantly higher or lower than the rest of the values. When plotted, an outlier appears as an isolated dot far away from the main cluster; these values can significantly shift the Mean but usually have little to no effect on the Median.

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Data Distribution describes how data is spread across its range. In a perfectly symmetrical distribution, the mean, median, and mode are all located at the same central point, creating a balanced 'mirror-image' shape on either side of the center.

📐Formulae

Mean(xˉ)=∑xn\text{Mean} (\bar{x}) = \frac{\sum x}{n}

Median Position=n+12\text{Median Position} = \frac{n + 1}{2}

Range=Maximum Value−Minimum Value\text{Range} = \text{Maximum Value} - \text{Minimum Value}

Mean of Even Median=Value at (n2)+Value at (n2+1)2\text{Mean of Even Median} = \frac{\text{Value at } (\frac{n}{2}) + \text{Value at } (\frac{n}{2} + 1)}{2}

💡Examples

Problem 1:

Calculate the Mean, Median, Mode, and Range for the following test scores: 85,92,78,85,9085, 92, 78, 85, 90.

Solution:

  1. Order the data: 78,85,85,90,9278, 85, 85, 90, 92
  2. Mean: 78+85+85+90+925=4305=86\frac{78 + 85 + 85 + 90 + 92}{5} = \frac{430}{5} = 86
  3. Median: The middle value in the sorted list (n=5n=5) is the 3rd term: 8585
  4. Mode: The value 8585 appears most frequently (twice).
  5. Range: 92−78=1492 - 78 = 14

Explanation:

First, we arrange the data to easily find the median and mode. The mean is the total sum divided by the number of students. The range shows the difference between the highest and lowest scores.

Problem 2:

Find the median of the following set of temperatures: 12∘C,15∘C,10∘C,18∘C,22∘C,14∘C12^{\circ}C, 15^{\circ}C, 10^{\circ}C, 18^{\circ}C, 22^{\circ}C, 14^{\circ}C.

Solution:

  1. Order the data: 10,12,14,15,18,2210, 12, 14, 15, 18, 22
  2. Identify the count: n=6n = 6 (even number).
  3. Find the middle two positions: The middle terms are the 3rd and 4th values: 1414 and 1515.
  4. Calculate Median: 14+152=14.5∘C\frac{14 + 15}{2} = 14.5^{\circ}C

Explanation:

Because there is an even number of data points, there is no single middle number. We take the mean of the two central values in the ordered list to find the median.