Collection and presentation of data

Primary vs secondary data. Primary data is collected first-hand by the person doing the study — a student polling classmates on their favourite sport. Secondary data has already been collected and published by someone else and is simply reused — pulling numbers from a government census report. Knowing which kind you are handling matters because primary data lets you control how the question was asked, while secondary data inherits whatever choices the original collector made.

Raw data and the frequency distribution table. Data fresh off a survey or test, before any organising, is called raw or ungrouped data — a plain, usually unordered, list. To make it readable you build a frequency distribution table: list every distinct value that appears, and record how many times it occurs, its frequency, written $f$. Tally marks (groups of four strokes with a fifth struck through) are the standard bookkeeping trick for counting occurrences in a single pass through the raw list without losing count.

The three summary numbers. Once the frequency table exists, three single numbers describe the whole data set from different angles:

• The mean $\bar{x}$ is the arithmetic average — every observation contributes equally.
• The median is the value that sits in the middle once the data is sorted — it is not pulled around by one extreme outlier the way the mean can be.
• The mode is simply whichever value shows up most often — useful when you care about the single most common outcome, such as the most common shoe size sold.

For raw (ungrouped) data of $n$ observations $x_1,x_2,\dots,x_n$, the mean is

where: $x_i$ is the $i$-th observation and $n$ is the total count of observations. The median needs the data sorted first, ascending or descending. If $n$ is odd, the median sits at position $\frac{n+1}{2}$ in the sorted list. If $n$ is even, there are two middle positions, $\frac{n}{2}$ and $\frac{n}{2}+1$, and the median is their average. The mode is read straight off the frequency table: whichever value has the largest $f$. A data set can have exactly one mode, more than one mode (if two or more values tie for highest frequency), or no mode at all (if every value occurs equally often).

The goal is to show why summing $f_i x_i$ across the distinct values and dividing by the total count gives exactly the same mean as adding up every raw observation one by one — so that once data is tallied, you never need to revisit the original unsorted list to find the average.

Step 1. Start from the raw definition. With $n$ raw observations $x_1,\dots,x_n$, the mean is $\bar{x} = \dfrac{\sum x_i}{n}$, adding every single value once.

Step 2. Group the identical values. Suppose a distinct value $v_1$ appears $f_1$ times, $v_2$ appears $f_2$ times, and so on up to $v_k$ appearing $f_k$ times, with $f_1+f_2+\cdots+f_k = n$. Adding $v_1$ to itself $f_1$ times is the same as computing $f_1 v_1$, so the full sum of all $n$ raw observations can be rewritten as $f_1v_1 + f_2v_2 + \cdots + f_kv_k = \sum f_i v_i$.

Step 3. Substitute this grouped sum back into the mean formula. Since $\sum x_i = \sum f_i v_i$ and $n = \sum f_i$, the mean becomes $\bar{x} = \dfrac{\sum f_iv_i}{\sum f_i}$ — the frequency-table formula, built from nothing more than regrouping equal terms in an addition.

Step 4. Read the result. No information is lost in tallying: multiplying each distinct value by how often it occurs, summing those products, and dividing by the total count of observations gives the identical mean you would get by patiently adding every one of the $n$ raw numbers.

A survey of 20 households recorded the number of family members in each: $3, 5, 4, 3, 2, 4, 4, 5, 3, 6, 4, 5, 3, 4, 2, 6, 4, 3, 5, 4$. Build the frequency distribution table and find the mean, median and mode of family size.

Step 1. Tally the raw list into a frequency table. Family size 2 occurs 2 times, 3 occurs 5 times, 4 occurs 7 times, 5 occurs 4 times, and 6 occurs 2 times. Check: $2+5+7+4+2 = 20$, matching the 20 households, so no value was missed or double-counted.

Step 2. Compute the mean using $\bar{x} = \dfrac{\sum f_iv_i}{\sum f_i}$: $\sum f_iv_i = (2\times2)+(5\times3)+(7\times4)+(4\times5)+(2\times6) = 4+15+28+20+12 = 79$, so $\bar{x} = \dfrac{79}{20} = 3.95$.

Step 3. Find the median by sorting all 20 values in ascending order: $2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,6,6$. Here $n=20$ is even, so average the values at positions $\frac{20}{2}=10$ and $\frac{20}{2}+1=11$. Both the 10th and 11th values are $4$, so the median is $4$.

Step 4. Find the mode by reading the highest frequency in the table: family size 4 occurs 7 times, more than any other value, so the mode is $4$.

The 20 households have mean family size $3.95$, median $4$, and mode $4$ — all three summaries agree the "typical" household has around 4 members, with the mean pulled down slightly by the smaller households of size 2.

• Confusing the frequency with the observation itself. In a table row "4 | 7", the 4 is the value and 7 is how many times it occurred — averaging the frequency column instead of the value column gives nonsense.
• Finding the median without sorting first. The position rule ($\frac{n+1}{2}$ or the two middle positions) only works on data already arranged in ascending or descending order.
• Picking one middle value when $n$ is even. Even-count data has two middle positions; the median is their average, not just one of them.
• Assuming a mode always exists and is unique. Data can be bimodal (two tied values), or have no mode at all if every value occurs equally often.

Ask the tutor: Want to build your own frequency table from a jumbled list and check your mean, median and mode step by step? Ask the AI tutor for a fresh practice data set.