6.1 Averages, Totals & Simple Statistics

6.1 Averages, Totals & Simple Statistics

Quick Answer: For entrance-exam mathematics, mean = sum ÷ count, median = middle value after sorting, mode = most frequent value, and range = highest − lowest. Weighted averages multiply each group by its size (or weight) before dividing. Always recover the total as a check: mean × count must equal the sum of the values.

The Service Commissions Department (SCD) lists Mathematics as one of the three multiple-choice subjects in its published entrance-examination notices. Applied arithmetic—especially averages, totals, and simple statistics—is classic civil-service exam material. You do not need advanced statistics; you need clean procedures, careful reading, and arithmetic you can re-check. This section uses everyday Trinidad and Tobago situations (shopping, transport fares, household water, shift hours). It does not teach firefighting hydraulics or technical fireground math.

Why totals come before averages

Many candidates jump straight to “add and divide” and lose points on careless totals. Build a habit:

1. List every value the question gives (and only those values).
2. Add carefully to get a total (sum).
3. Count the number of values n.
4. Compute the mean: mean = total ÷ n.
5. Sanity-check: mean × n should return the total (allowing for intentional rounding).

Worked example — market shopping total and mean

A family in San Fernando buys vegetables for five days:

Total = 48 + 36 + 52 + 40 + 54.

Step: 48 + 36 = 84; 84 + 52 = 136; 136 + 40 = 176; 176 + 54 = 230.

Mean daily spend = 230 ÷ 5 = 46 TTD.

Check: 46 × 5 = 230. The check matches, so the mean is reliable.

Including a missing day

Suppose Saturday’s spend is missing, and the six-day mean is said to be 50 TTD. Then the six-day total must be 50 × 6 = 300. Days Mon–Fri total 230, so Saturday = 300 − 230 = 70 TTD. This “find the missing value from the mean” pattern is very common.

Mean (arithmetic average)

The arithmetic mean treats every observation as equally important unless the question says otherwise.

Formula:
[ \text{Mean} = \frac{x_1 + x_2 + \cdots + x_n}{n} ]

Example — bus-route passenger counts

A maxi-taxi driver on a Port of Spain–Arima run records passengers over four trips: 14, 18, 11, 17.

Total = 14 + 18 + 11 + 17 = 60.
Mean = 60 ÷ 4 = 15 passengers per trip.

If a fifth trip carries 20 passengers, the new total is 80 and the new mean is 80 ÷ 5 = 16. Adding a value above the old mean pulls the mean up; adding a value below pulls it down. That qualitative check helps eliminate wrong options quickly.

Median — the middle after sorting

The median resists extreme outliers better than the mean. Procedure:

1. Sort the data from smallest to largest.
2. If n is odd, the median is the middle term at position ((n+1)/2).
3. If n is even, the median is the mean of the two middle terms.

Odd count example

Ages of five household members (years): 9, 42, 15, 38, 11.

Sorted: 9, 11, 15, 38, 42.
Middle (3rd of 5) = 15. Median age = 15 years.

Mean would be (9+11+15+38+42) ÷ 5 = 115 ÷ 5 = 23, higher because of the adult ages. Exam questions sometimes ask which measure is “most affected by extreme values”—the mean.

Even count example

Weekly hours a student spends on homework: 6, 4, 9, 7.

Sorted: 4, 6, 7, 9.
Two middle values: 6 and 7. Median = (6 + 7) ÷ 2 = 6.5 hours.

Mode and range

• Mode: the value that appears most often. A data set can have one mode, more than one mode, or no mode (all frequencies equal).
• Range: maximum − minimum. It measures spread in the simplest way.

Combined example

Quiz scores out of 20: 12, 15, 12, 18, 12, 16.

Sorted: 12, 12, 12, 15, 16, 18.

Always read the question: “most common,” “middle,” “average,” and “difference between highest and lowest” map to mode, median, mean, and range respectively.

Weighted totals and weighted averages

When groups are unequal, a plain mean of group averages is wrong. Use weights.

Weighted average = (\dfrac{\sum (\text{value} \times \text{weight})}{\sum \text{weights}})

Example — two-week wages

A shop assistant earns 120 TTD/day for 5 days in week 1 and 150 TTD/day for 3 days in week 2. What is the average daily pay across all days worked?

Incorrect shortcut: (120 + 150) ÷ 2 = 135 — wrong, because the weeks have different numbers of days.

Correct:
Total pay = (120 × 5) + (150 × 3) = 600 + 450 = 1,050 TTD.
Total days = 5 + 3 = 8.
Average daily pay = 1,050 ÷ 8 = 131.25 TTD.

Example — average speed over unequal distances (preview of §6.4)

If you drive 60 km at 40 km/h and 40 km at 60 km/h, you cannot average 40 and 60 to get 50. Weighted (by time) methods are required; speed-distance-time is covered fully in section 6.4. For now, remember: unequal weights demand weighted methods.

Example — class average from group sizes

Form 5A has 20 students with mean score 65. Form 5B has 30 students with mean score 75. Combined mean?

Total points A = 65 × 20 = 1,300.
Total points B = 75 × 30 = 2,250.
Combined total = 3,550. Combined students = 50.
Combined mean = 3,550 ÷ 50 = 71.

Trap: averaging 65 and 75 gives 70—close but incorrect because B is larger.

Building multi-step total questions

Entrance items often chain ideas:

1. Find several subtotals.
2. Combine into a grand total.
3. Divide for a mean or a unit rate.
4. Sometimes subtract a discount or add tax (percent skills from Chapter 5).

Example — household water tank (civilian use only)

A rooftop tank holds 1,200 litres. Over three days the household uses 280 L, 310 L, and 250 L. Rain refill adds 150 L on day 2 only. How many litres remain after three days if the tank started full?

Day 1: 1,200 − 280 = 920.
Day 2: 920 − 310 + 150 = 760.
Day 3: 760 − 250 = 510 litres remaining.

Mean daily net change is not required unless asked—but if asked, total change from full is 1,200 − 510 = 690 used net of refill, and you must state what “average use” means carefully (gross use vs net after refill). Precision in reading prevents wrong options.

Simple frequency tables

Data may appear in a frequency table rather than a long list.

Total of scores = Σ(x × f) = (2×3) + (3×5) + (4×2) = 6 + 15 + 8 = 29.
Number of observations n = Σf = 3 + 5 + 2 = 10.
Mean = 29 ÷ 10 = 2.9.

Median position for n = 10 is average of 5th and 6th ordered values. Ordered: three 2s, five 3s, two 4s → 5th and 6th are both 3, so median = 3.

Exam traps and good habits

Estimation check

If five numbers are all near 40, the mean should be near 40. A calculated mean of 400 almost always means a decimal-place or total error. Use magnitude checks before selecting an option.

Connecting averages to other arithmetic

Averages sit on top of Chapter 5 skills:

• Percentages: “average increased by 10%” → new mean = old mean × 1.10.
• Ratios: parts of a total can be averaged after converting parts to actual amounts.
• Fractions: mean of mixed numbers needs careful conversion to improper fractions or decimals.

Example — percent change of a mean

Weekly mean grocery spend was 200 TTD. Next month it rises by 15%. New mean = 200 × 1.15 = 230 TTD. If there are 4 weeks, monthly total ≈ 230 × 4 = 920 TTD (if every week equals the new mean).

Strategy for MCQ timing

1. Underline whether the question wants mean, median, mode, range, total, or missing value.
2. Write the total on scratch paper before dividing.
3. For weighted items, draw a two-column table: value | weight.
4. Eliminate options that ignore weights or forget to sort.
5. Re-multiply mean × n as a 5-second final check when time allows.

Summary checklist before the next section

• I can compute mean, median, mode, and range from a short list.
• I can find a missing value when the mean and other values are known.
• I can compute a weighted average for unequal groups.
• I can read a simple frequency table and find Σ(x f).
• I re-check with mean × count = total.

Section 6.2 moves from pure number averages to measurement units and conversions—essential when totals mix metres with kilometres or litres with millilitres. Keep the same “total first, then divide” discipline when units change mid-problem.