4.3 Descriptive Statistics and Data Interpretation

4.3 Descriptive Statistics and Data Interpretation

Candidate reports describe the AdUCET mathematics items as leaning unusually hard on data handling: a frequency table to average, a graph to read, a median to locate, a claim about spread to judge. These are among the fastest marks available because the arithmetic stays small — but only if you can tell instantly which summary the item is asking for. This section builds that reflex, from the three kinds of mean through the five-number summary to the shape of a distribution.

1. Measures of Centre

1.1 The mean and its three faces

The arithmetic mean is the total divided by the count:

$\bar{x} = \frac{\sum x_i}{n}$

Two variants dominate entrance-test items.

• Weighted mean — used when values carry unequal importance, exactly as in a DepEd class record where written work, performance tasks and the quarterly assessment are worth different percentages: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$
• Grouped mean — used when the raw scores are gone and only a frequency table survives. Each class midpoint stands in for every score in that class: $\bar{x} = \frac{\sum f_i m_i}{\sum f_i}, \qquad m_i = \frac{\text{lower limit} + \text{upper limit}}{2}$

1.2 Median

The median is the middle value of ordered data, so sort first — always. With $n$ odd the median is the value in position $(n+1)/2$. With $n$ even it is the mean of the values in positions $n/2$ and $n/2 + 1$. For six monthly water bills of ₱285, ₱295, ₱320, ₱350, ₱380 and ₱410, the median is $(320 + 350)/2 = ₱335$ — a figure that is not itself in the list, which is perfectly normal.

1.3 Mode and modality

The mode is the value that occurs most often. A data set is unimodal (one clear mode), bimodal (two values tie for the top frequency), multimodal, or has no mode when every value appears equally often. The mode is the only measure of centre that works on categorical data — favourite jeepney route, blood type, senior high strand.

1.4 Choosing the right summary

1.5 Outliers move the mean, not the median

Five students report daily allowances of ₱100, ₱120, ₱150, ₱150 and ₱180. The total is ₱700, so the mean is ₱140, the median is ₱150 and the mode is ₱150. Add one classmate whose allowance is ₱900: the total becomes ₱1,600 and the mean leaps to $1600/6 \approx ₱266.67$, a figure larger than five of the six actual allowances. The median only moves from ₱150 to $(150 + 150)/2 = ₱150$ — it does not move at all. This single fact is why newspapers quote median salaries and median household income.

2. Worked Example 1 — a frequency table from end to end

Problem. Forty Grade 12 students sat a 40-item mathematics review test. Find the mean, the modal class, the median class, and the central angle that the class 26–30 would occupy in a pie chart.

1. Midpoints. $m_1 = (11 + 15)/2 = 13$; every later midpoint adds the class width of 5.
2. Mean. $\bar{x} = 1080 / 40 = 27.0$ marks. Note that 27 is not the average of the midpoints (25.5); the frequencies pull it upward.
3. Modal class. The largest frequency is 12, so the modal class is 26–30.
4. Median class. With $n = 40$ the median sits between the 20th and 21st ordered scores. The cumulative frequency reaches only 16 by the end of 21–25 but 28 by the end of 26–30, so both scores lie in 26–30.
5. Pie-chart angle. Relative frequency of 26–30 is $12/40 = 0.30$, so the central angle is $0.30 \times 360^\circ = 108^\circ$.

3. Measures of Spread

• Range $= x_{\max} - x_{\min}$. Simple, but a single extreme value destroys it.
• Quartiles. $Q_1$ is the median of the lower half, $Q_2$ is the median itself, $Q_3$ is the median of the upper half. The interquartile range is $\text{IQR} = Q_3 - Q_1$ — the width of the middle 50% of the data. A value below $Q_1 - 1.5(\text{IQR})$ or above $Q_3 + 1.5(\text{IQR})$ is conventionally flagged as an outlier.
• Variance is the mean of the squared deviations from the mean; standard deviation is its positive square root and is expressed in the original unit. $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \quad\text{(population)} \qquad s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \quad\text{(sample)}$

Divide by $N$ when your data is the entire group you care about (all 40 students in one section). Divide by $n - 1$ when the data is a sample drawn from a larger population. Because $n - 1 < n$, the sample figure is always the larger of the two.

4. Worked Example 2 — standard deviation by hand

Problem. Five students score 6, 8, 9, 10 and 12 on a 15-item quiz. Find the population and sample standard deviations.

1. Mean. $6 + 8 + 9 + 10 + 12 = 45$, so $\bar{x} = 45/5 = 9$.
2. Deviations. $-3,; -1,; 0,; +1,; +3$. They sum to zero, as deviations always must — a free accuracy check.
3. Squared deviations. $9,; 1,; 0,; 1,; 9$, whose sum is 20.
4. Population. $\sigma^2 = 20/5 = 4$, so $\sigma = 2$ marks.
5. Sample. $s^2 = 20/4 = 5$, so $s = \sqrt{5} \approx 2.24$ marks.

Shifting and scaling

So awarding all five students a 5-point bonus moves the mean from 9 to 14 and leaves $\sigma = 2$: every score shifted, none spread out. Doubling every score gives a mean of 18, a standard deviation of 4 and a variance of 16.

5. Shape: Skewness and the Empirical Rule

The mnemonic is simply that the mean chases the tail: whichever side the long tail points to, the mean is dragged that way past the median.

  Right-skewed (positive)              Left-skewed (negative)
     ^                                    ^
     |  .-.                               |            .-.
     | /   \__                            |        __/   \
     |/       \____                       |   ____/       \|
     +-------------------->               +-------------------->
      Mo Md  Mean                            Mean Md Mo

For a roughly bell-shaped distribution the empirical rule (the 68-95-99.7 rule) says about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. If a mock exam has mean 78 and standard deviation 6, then about 68% of scores lie between 72 and 84, about 95% between 66 and 90, and about 99.7% between 60 and 96. Because the curve is symmetric, the share above 84 is about $(100 - 68)/2 = 16%$.

6. Reading the Standard Displays

For the ordered data 12, 15, 17, 18, 21, 24, 26, 30 the median is $(18 + 21)/2 = 19.5$, $Q_1 = (15 + 17)/2 = 16$, $Q_3 = (24 + 26)/2 = 25$ and the IQR is 9. The fences sit at $16 - 13.5 = 2.5$ and $25 + 13.5 = 38.5$, so nothing here is an outlier.

   min      Q1     Md      Q3        max
    12      16    19.5     25         30
    |--------[=====|========]----------|
   10   14   18   22   26   30   34

The box holds the middle 50%; a box that is longer on the right with a longer right whisker signals right skew.

7. Common Traps

1. Not sorting before taking a median. The single most common careless error in this topic.
2. Taking one middle value when $n$ is even. With 6 values you must average the 3rd and 4th, not pick one.
3. Averaging the frequency column. In a frequency table the frequencies are weights, not data values.
4. Averaging the midpoints instead of weighting them. In Worked Example 1 that would give 25.5 instead of the correct 27.
5. Using $n$ when the item says "a sample of". The divisor for a sample variance is $n - 1$.
6. Assuming a bonus changes the spread. Adding a constant shifts the centre and leaves every deviation identical, so the standard deviation is untouched.
7. Confusing percentage with angle in a pie chart. A 30% slice is $108^\circ$, not $30^\circ$.
8. Reversing the skew ordering. In a right-skewed set the mean is the largest, because it is the measure the outliers can reach.