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Statistics and Probability - Measures of central tendency for grouped and ungrouped data (mean, median, mode)

Grade 9IB

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. In a visual representation like a dot plot, the central tendency indicates where most of the data points cluster together.

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The Mean (Arithmetic Average) is calculated by summing all data values and dividing by the total count. For grouped data, we estimate the mean by assuming all values in an interval occur at the mid-interval value, which acts as the balance point for that specific group.

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The Median is the middle value when data is arranged in ascending order. If there is an even number of observations, it is the average of the two middle terms. In a histogram, the median is the value that divides the total area of the bars into two equal halves.

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The Mode is the most frequently occurring value in a dataset. In a frequency table or bar chart, the mode is easily identified as the category with the highest frequency or the tallest bar. A dataset can be bimodal (two modes) or have no mode if all values appear with equal frequency.

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Grouped Data involves organizing data into intervals or classes (e.g., 10≤x<2010 \le x < 20). Because individual data points are lost, we use the 'Modal Class' (the interval with the highest frequency) and the 'Mid-interval value' to perform calculations.

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Mid-interval values are essential for calculating the mean of grouped data. You find the midpoint by calculating the average of the lower and upper boundaries of the class. Visually, this is the center point on the x-axis for each bar in a frequency histogram.

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The choice of measure depends on the data distribution. The mean is sensitive to outliers (extreme values that sit far away from the main cluster on a number line), whereas the median is more robust and provides a better 'middle' representation for skewed data.

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Cumulative Frequency is often used to find the median for grouped data. By plotting a cumulative frequency curve (an S-shaped curve or ogive), the median is the x-value corresponding to the 50% mark on the y-axis.

📐Formulae

Mean for ungrouped data: xˉ=∑xn\bar{x} = \frac{\sum x}{n}

Mean for grouped data: xˉ=∑(f⋅x)∑f\bar{x} = \frac{\sum (f \cdot x)}{\sum f} (where xx is the mid-interval value and ff is the frequency)

Position of the median in a sorted list of nn items: Position=n+12\text{Position} = \frac{n + 1}{2}

Mid-interval value: xmid=Lower Bound+Upper Bound2x_{mid} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}

Total frequency: n=∑fn = \sum f

💡Examples

Problem 1:

Find the mean, median, and mode for the following set of test scores: 12,15,12,18,2012, 15, 12, 18, 20.

Solution:

  1. Mean: Sum the values: 12+15+12+18+20=7712 + 15 + 12 + 18 + 20 = 77. Divide by the count (n=5n=5): xˉ=775=15.4\bar{x} = \frac{77}{5} = 15.4. \n2. Median: Sort the data: 12,12,15,18,2012, 12, 15, 18, 20. The middle value (3rd position) is 1515. \n3. Mode: The value 1212 appears twice, while others appear once. Mode = 1212.

Explanation:

This example demonstrates central tendency for a small, ungrouped dataset. The mean provides the average, the median shows the central score after ordering, and the mode identifies the most common score.

Problem 2:

Calculate the estimated mean for the following grouped frequency table: \n- Weight 40≤w<5040 \le w < 50: Frequency 33 \n- Weight 50≤w<6050 \le w < 60: Frequency 55 \n- Weight 60≤w<7060 \le w < 70: Frequency 22

Solution:

  1. Find Mid-intervals (xx): \n - For 40−5040-50: x=40+502=45x = \frac{40+50}{2} = 45 \n - For 50−6050-60: x=50+602=55x = \frac{50+60}{2} = 55 \n - For 60−7060-70: x=60+702=65x = \frac{60+70}{2} = 65 \n2. Calculate f⋅xf \cdot x: \n - 3×45=1353 \times 45 = 135 \n - 5×55=2755 \times 55 = 275 \n - 2×65=1302 \times 65 = 130 \n3. Sum the totals: \n - ∑f=3+5+2=10\sum f = 3 + 5 + 2 = 10 \n - ∑(f⋅x)=135+275+130=540\sum (f \cdot x) = 135 + 275 + 130 = 540 \n4. Calculate Mean: \n - xˉ=54010=54\bar{x} = \frac{540}{10} = 54 kg.

Explanation:

For grouped data, we use the midpoint of each class as a representative value for that group to estimate the sum of all values before dividing by the total frequency.