Measures of Central Tendency Overview
📌 Measures of central tendency provide a representative value for a data sample that will subsequently represent the population.
📊 These measures are divided into three main types: Mean, Median, and Mode.
🧠 The video series previously covered creating grouped frequency distribution tables.

Calculating the Mean (Average)
📐 The Mean ($\bar{x}$) is calculated by dividing the sum of all data values by the total number of data points ($n$): $\bar{x} = \frac{\sum x_i}{n}$.
📊 For weighted data distributions, the formula is $\bar{x} = \frac{\sum f \cdot x}{\sum f}$, where $f$ is the frequency.
➕ Example Calculation (Ungrouped Data): For data $\{2, 5, 2, 1, 6, 3, 3, 3\}$, the mean is $24 / 8 = 3$.
📈 Example Calculation (Weighted Data): A table yielded a mean of $44 / 10 = 4.4$.

Solving Mean Problems with Combined Groups
🔗 To find the total mean when given means for subgroups, use the weighted average approach.
🚹 Example: For 8 male students with a mean score of 60 and 12 female students with a mean score of 75, the total mean is $\frac{(8 \times 60) + (12 \times 75)}{8 + 12} = \frac{480 + 900}{20} = \frac{1380}{20} = 69$. (Note: Calculation in transcript showed $70$ for female mean, resulting in $66$ for the answer: $\frac{480+840}{20} = \frac{1320}{20} = 66$).
🆕 Example: If 9 students have a mean score of 7, and a new student with a score of 6 joins, the new mean for 10 students is $\frac{(9 \times 7) + 6}{9 + 1} = \frac{63 + 6}{10} = 6.9$.

Calculating the Median
❤️ The Median is the middle value in a dataset after it has been ordered from smallest to largest.
✂️ To find the middle value, systematically cross out one value from the left end and one from the right end until only the center value(s) remain.
🧮 If two middle values exist (for an even number of data points), the median is the average of those two numbers: $\frac{\text{Value}_1 + \text{Value}_2}{2}$. For data points $2$ and $3$ in the middle, the median is $(2 + 3) / 2 = 2.5$.

Identifying the Mode
🌟 The Mode is the data value that appears most frequently in the set.
🔁 For ungrouped data, compare the frequency of each distinct number; the number(s) with the highest count is the mode.
🔢 Example: In data where '1' appears three times and '2' appears three times (the maximum frequency), the modes are 1 and 2.
📊 For weighted distribution tables, simply identify the $x$ value corresponding to the highest frequency ($f$).

Key Points & Insights
➡️ The fundamental task is to identify which of the three measures (Mean, Median, Mode) is required for the problem.
➡️ When calculating the Mean, clearly distinguish between raw data sums and weighted sums ($\sum f \cdot x$).
➡️ For the Median, the crucial first step is always ordering the data before locating the center point(s).
➡️ The Mode relies solely on observing the highest frequency in the dataset or frequency table.

📸 Video summarized with SummaryTube.com on Jan 14, 2026, 03:39 UTC