Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Count and organize repeated values.
- Compute mean and median correctly.
- Interpret a relative frequency without overstating it.
Before you start
Addition, division, fractions, and ordering whole numbers.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Eight recorded bus delays, in minutes, are 0, 1, 1, 2, 2, 2, 3, 5. What is a useful numerical summary, and what fraction were at most two minutes late?
Why this math matters
A list of observations becomes easier to understand when we separate individual measurements, repeated values, and summaries. Different summaries answer different questions: the mean balances the total, the median locates the middle, and a relative frequency describes how often a condition occurred in these observations. None automatically predicts every future trip.
Set up the model
A useful answer starts with clear assumptions:
- Each value belongs to one different arriving bus in this fictional record.
- All delays use the same minute unit and recording convention.
- The eight buses are not claimed to be a random or representative sample of all buses.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
What do eight bus-delay observations actually tell us?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Eight recorded bus delays, in minutes, are 0, 1, 1, 2, 2, 2, 3, 5. What is a useful numerical summary, and what fraction were at most two minutes late?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
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Check the frequencies
0: 1; 1: 2; 2: 3; 3: 1; 5: 1; total = 8
The frequencies add back to the observation count. This catches missing values and shows that a two-minute delay is the most common recorded value.
Find two measures of center
mean = 16/8 = 2 min; median = (2 + 2)/2 = 2 min
The list is already ordered. With eight observations, average the fourth and fifth values for the median. The two centers happen to agree here.
Answer the threshold question
delays ≤ 2: 6 buses; relative frequency = 6/8 = 75%
Include the buses delayed by exactly two minutes because the question says at most two.
The result
The mean and median are both 2 minutes, and 75% of these recorded buses were delayed by at most 2 minutes.
The five-minute delay affects the mean, but a single summary hides the distribution. Report the count and context alongside percentages. A prediction about all future buses would require a defensible sampling or probability model.
Common mistakes to catch
- At most two includes exactly two; less than two would count only three buses.
- A 75% observed fraction from this small record is not a guaranteed future arrival probability.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What fraction of these buses were delayed by more than two minutes?
Show a hint
Count the observations 3 and 5.
Reveal answer and explanation
2/8 = 25%
Two of the eight values exceed two. This complements the 75% at or below two.
Practice 2
Find the mean and median of delays 1, 1, 2, 8.
Show a hint
Sort first, then average the two middle values for the median.
Reveal answer and explanation
Mean 3 minutes; median 1.5 minutes
The total is twelve, so the mean is three. The middle values one and two average to 1.5.
Take the idea with you
Before summarizing data, state what one observation represents, its unit, and how the observations were selected.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Can two processes have the same average but different reliability?
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