11.1 Measures of Central Tendency

11.1 Measures of Central Tendency

Quick Answer: The three classic measures of central tendency are the mean (arithmetic average), the median (middle value of ordered data), and the mode (most frequent value). USTET Statistics items test whether you can compute each correctly, choose the right one for a story problem, and recognize how a single extreme score changes the picture.

Statistics and data analysis sit inside the USTET Mathematics subtest alongside algebra, geometry, and General Mathematics. You will not need advanced college statistics — no standard-deviation formulas from a formula sheet, no hypothesis tests — but you will need fluent Grade 11 skill with averages, middle values, and most-common values under timed multiple-choice pressure. This section builds that fluency.

Why Central Tendency Matters on USTET

Entrance-exam writers love central-tendency questions because they look short but punish careless reading. A stem may give five quiz scores and ask for the mean; the next item may add one very high makeup score and ask which measure changed the most. Another item may present a frequency table of ages and ask for the modal class. If you memorize only “average = add and divide,” you will miss median and mode items and stumble when the data set has an even count or repeated values.

Treat every data set the same way on test day:

1. Copy the numbers onto scratch paper (do not compute from the booklet alone if the list is long).
2. Count how many values you have — that count is your mean’s denominator and tells you whether the median uses one middle value or two.
3. Order the list from least to greatest before finding the median.
4. Scan for repeats before naming the mode.
5. Ask what the question wants — average, middle, most common, or “least affected by an outlier.”

The Mean (Arithmetic Average)

For a list of numbers $x_1, x_2, \ldots, x_n$, the mean (often written $\bar{x}$ or $\mu$ in textbooks) is

$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum x}{n}.$

Worked example. Ana’s USTET practice Math scores for five drills are 72, 85, 90, 68, and 85. The sum is $72 + 85 + 90 + 68 + 85 = 400$. The mean is $400 \div 5 = 80$.

Common mean traps:

• Wrong divisor. Using 4 instead of 5 after forgetting a score, or dividing by the largest value instead of the count.
• Including labels. Adding category numbers that are not data values (for example, treating “Item 1, Item 2…” as scores).
• Mixing units. Averaging pesos and counts in the same sum without converting — rare on USTET, but word problems sometimes hide an extra quantity you should not include.

Weighted Mean

Sometimes values do not count equally. If a teacher says quizzes are 40% and the final exam is 60%, the class average is a weighted mean:

$\bar{x}_w = \frac{\sum (w_i \cdot x_i)}{\sum w_i}.$

Worked example. Carlo scores 88 on quizzes (weight 0.40) and 76 on the final (weight 0.60). Weighted mean: $(0.40)(88) + (0.60)(76) = 35.2 + 45.6 = 80.8$.

If weights are given as frequencies in a table (for example, 3 students scored 70, 5 scored 80, 2 scored 90), treat frequency as the weight: multiply each score by its frequency, add the products, and divide by the total number of students.

Mean $= 790 \div 10 = 79$.

The Median

The median is the middle value once the data are arranged in order.

• If $n$ is odd, the median is the value in position $\frac{n+1}{2}$.
• If $n$ is even, the median is the average of the two middle values (positions $\frac{n}{2}$ and $\frac{n}{2}+1$).

Odd count. Ordered scores: 68, 72, 85, 85, 90. Here $n = 5$, middle position $= 3$, median $= 85$.

Even count. Ordered scores: 68, 72, 85, 90. Here $n = 4$, middle pair $= 72$ and $85$, median $= (72 + 85) \div 2 = 78.5$.

Never find the median from an unordered list. The number sitting in the middle of a messy stem is usually not the median.

The Mode

The mode is the value that occurs most frequently.

• Unimodal: one mode (example: 85 appears twice above; all others once → mode 85).
• Bimodal: two values tie for highest frequency.
• Multimodal: more than two modes.
• No mode: every value appears equally often (often once each).

For categorical data (favorite subject, blood type, barangay), the mode is often the only sensible central measure — you cannot meaningfully average “Math” and “Science.”

Choosing Mean vs Median vs Mode

Outlier effect. Start with 10, 12, 14, 16, 18. Mean $= 14$, median $= 14$, no unique mode. Replace 18 with 80: new list 10, 12, 14, 16, 80. Mean jumps to $26.4$, median stays $14$, still no unique mode. USTET loves this contrast: the mean moved; the median did not.

Frequency Tables and Grouped Hints

Grade 11 problems may give a simple frequency table rather than a raw list. Expand mentally or on paper: if “score 75” has frequency 4, that contributes four 75s to the data set. For the median of a large frequency table, find the cumulative frequency and locate the middle position — you do not need to write every repeated value if you track running totals carefully.

Grouped-data medians (class intervals like 70–79) appear less often on entrance exams than raw lists, but if you see class intervals, identify the modal class (interval with highest frequency) when asked for the mode of grouped data.

Speed Habits for the Math Subtest

• Estimate first: if scores are all near 80, a mean of 45 is almost certainly a calculation error.
• When options are close (79 vs 80 vs 81), re-add the sum before dividing.
• For even $n$, do not forget to average the two middle numbers — picking only the lower or upper middle value is a classic wrong answer planted in choices.
• If a stem says “which measure best represents…,” look for outlier language (“one student scored much higher,” “including a bonus outlier”).

Master mean, median, and mode as a trio. USTET will mix computation items with judgment items; both reward the same careful ordering, counting, and choice of measure.