Calculating Mean, Median, Mode, and Midrange from a Frequency Distribution Table

Math Problem Statement

Here is a FDT for a small data set:

datafreq

Find the following measures of central tendency.

mean =

median =

mode =

midrange =

Solution

Let's compute the measures of central tendency based on the provided data frequency table. The data and their frequencies are as follows:

Data ( $x$ )	Frequency ( $f$ )
25	1
26	4
27	3
28	3
29	9

1. Mean ( $\bar{x}$ )

The mean is calculated using the formula: $\bar{x} = \frac{\sum (x \cdot f)}{\sum f}$

Steps:

Compute $x \cdot f$ for each value of $x$ .
Find the sum of $x \cdot f$ and the sum of $f$ .
Divide the sum of $x \cdot f$ by the sum of $f$ .

&\sum f = 1 + 4 + 3 + 3 + 9 = 20, \\ &\sum (x \cdot f) = (25 \cdot 1) + (26 \cdot 4) + (27 \cdot 3) + (28 \cdot 3) + (29 \cdot 9).$$ ### 2. Median The median is the middle value of the data set when arranged in ascending order. For grouped data: - Compute the cumulative frequency to determine where the median lies. - Use the formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f_m}\right) \cdot h,$$ where: - $$L$$ is the lower boundary of the median class, - $$N$$ is the total frequency, - $$CF$$ is the cumulative frequency before the median class, - $$f_m$$ is the frequency of the median class, - $$h$$ is the class width. Since this data is ungrouped, the middle value is determined directly. ### 3. Mode The mode is the value with the highest frequency. ### 4. Midrange The midrange is calculated as: $$\text{Midrange} = \frac{\text{Minimum value} + \text{Maximum value}}{2}.$$ #### Solving: Let’s calculate step-by-step. Given data data = [25, 26, 27, 28, 29] # x values frequency = [1, 4, 3, 3, 9] # f values # Total frequency (sum of f) total_frequency = sum(frequency) # Weighted sum of data (sum of x * f) weighted_sum = sum(x * f for x, f in zip(data, frequency)) # Mean calculation mean = weighted_sum / total_frequency # Median calculation # Expand the dataset to find the middle value(s) expanded_data = [] for x, f in zip(data, frequency): expanded_data.extend([x] * f) expanded_data.sort() # Median depends on whether the total number of data points is odd or even n = len(expanded_data) if n % 2 == 1: median = expanded_data[n // 2] else: median = (expanded_data[n // 2 - 1] + expanded_data[n // 2]) / 2 # Mode calculation (value with the highest frequency) max_frequency = max(frequency) mode = [data[i] for i in range(len(data)) if frequency[i] == max_frequency] # Midrange calculation midrange = (min(data) + max(data)) / 2 mean, median, mode, midrange### Results: 1. **Mean**: $$27.75$$ 2. **Median**: $$28.0$$ 3. **Mode**: $$29$$ (it appears most frequently) 4. **Midrange**: $$27.0$$ Let me know if you'd like detailed explanations for any of these calculations. --- Here are five related questions to expand on this topic: 1. How would the mean and median change if the frequency of $$25$$ increased to $$5$$? 2. Can there be more than one mode in a data set? If so, provide an example. 3. How does the median relate to the mean in a skewed data set? 4. What is the impact of adding an outlier (e.g., $$35$$) to the dataset on the midrange? 5. How do grouped and ungrouped data differ in calculating these measures? **Tip:** The mean is sensitive to extreme values (outliers), while the median and mode are more robust measures of central tendency for skewed distributions.

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Math Problem Analysis

Mathematical Concepts

Statistics
Measures of Central Tendency

Formulas

Mean formula: (Σ(x * f)) / Σf
Median formula: L + ((N/2 - CF) / fm) * h (for grouped data)
Mode: The most frequent value in the data set
Midrange formula: (Min value + Max value) / 2

Theorems

Measures of Central Tendency

Suitable Grade Level

Grades 7-9

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