Unit 1 - Question Recognition

A course has weights of 5%, 10%, 15%, 25%, 45%. What mark is needed on the final?

or What is the overall course grade?

• "course grade"

• "overall mark"

• "worth"

• "weighted"

• "what mark is needed on the final"

Weighted Average (Grade) because you're taking several grades that have different weights and combining them into one overall course grade.

The reason we used fractions like:8/10, 70/100, 22/30, 28/35 is that these fractions tell us the proportion earned in each category before applying the weight.

Score / Total x Weight [for each category]

The following table shows the number of hours studied by students and the frequency of each value. Find the weighted mean number of hours studied.

Hours (x)	Frequency (f)
2	3
4	5
6	2

• Weighted mean

• Weighted average

• Frequency table

• Average with weights

• Average GPA using credit hours

Weighted Mean. The question directly says "weighted average" or "weighted mean."

• Clue: A frequency table is often a weighted mean question in disguise.

Two groups have different averages. The overall average is known. Find the missing number of students.

or

Find the missing group average.

The clues are:

• Group 1 average

• Group 2 average

• Overall average

• Number of people/items in each group

Weighted Mean (Weighted Average) Question with Group Averages.

• These are classic weighted mean questions because the groups have different sizes.

What are Attributes of Standard Deviation and Which Formula is Used?

SD measures the spread out the data are from the mean. When most values are near the centre, this indicates that there is a small distance from the mean. However, when most data is evenly spread everywhere, it means that there are lots of data spread out from the mean.

The standard deviation has the same units as the original observations.

• Example: Heights measured in cm, Standard deviation measured in cm

If the standard deviation is zero, then all observations have the same value.

• If every value is identical: x − xˉ = 0 for every observation.

The standard deviation should be used as the measure of variability when the mean is chosen as the measure of centre.

• Mean ↔ Standard deviation

• Median ↔ IQR

The standard deviation should not be used as a measure of spread when the distribution is strongly skewed.

For skewed data we prefer these because they are resistant to outliers:

• Median

• IQR

The standard deviation of the prices of ten used cars at a car dealership is calculated to be $2,570. If the dealership reduces the price of each car by $300, what is the value of the new standard deviation?

(A) $2,870 (B) $2,600 (C) $2,570 (D) $2,540 (E) $2,270

(A) $2,870 (B) $2,600 (C) $2,570 (D) $2,540 (E) $2,270

Standard deviation measures spread around the mean.

When every value moves down by the same amount:

• The mean moves down by $300.

• Every data point moves down by $300.

So the distances from the mean don't change. SD stays the same, because the actual spread itself is not changing by a variable amount.

Find the variance of the following sample: 2, 4, 6, 8

or

The variance of a data set is...

Trying to find the variance of the data.

A frequency table has missing RF values. Find the total number of observations.

What type of Question is This?

Finding the relative frequency.

Add up all the known relative frequencies and subtract that sum from

1.0

(if using decimals) or

100%

(if using percentages)

Look for:

• Relative Frequency

• Frequency Table

• Proportion

• Percent

Find the Median of the Ordered Dataset

What type of Question is This?

Look for:

• Median

• Middle value (Q2 or 50%)

• Ordered data

Find Q1​, Q3​, or the IQR.

What type of Question is This?

Finding the measures of positions within a dataset, that also appears in boxplots.

Keywords:

• Quartile

• Percentile

• IQR

• Five-number summary

Which values will be labeled as outliers on an outlier boxplot?

What type of Question is This?

This is looking at the outlier box plots. Trying to detect the potential outliers.

Keywords:

• Outlier

• Boxplot

• Suspected outlier

• Whiskers

What final grade is needed to earn 80% overall?

A course grade is based on:

• Assignments 20%

• Midterm 30%

• Final 50%

What type of Question is This?

This is a weighted grade question. Weight will add up to 100%.

Look for:

• Course grade

• Worth 20%

• Worth 50%

• Final exam needed

Section A has 123 students with an average of 70.2. Section B has an average of 62.8.

The overall average is 65.6. How many students are in Section B?

This is a type of weighted mean question. Whenever you see question with the average score, number of students, or combined average, this is a weighted mean.

Look for:

• Two groups

• Group averages

• Number of students

• Overall average

Which measures are resistant to outliers

Resistant Measures

✅ Median

✅ Quartiles

✅ IQR

These measures only depend on the position of values, not their exact size.

Not Resistant

❌ Mean

❌ Range

❌ Standard Deviation

These are not resistant because they use every actual value.

What graph should be used to display the distribution of quantitative data?

Histogram because the histogram shows the shape, centre, spread, skewness, clusters and gaps in the data

Use when data are:

✅ Quantitative

✅ Numerical

Examples:

• Age

• Height

• Weight

• Test Scores

• Income

Which graph should be used to track stock prices over time?

Timeplot should be used when data is collected overtime. This can be recognized through words like over time, month, year, day, trend.

Examples:

• Daily temperatures

• Stock prices

• Population growth

• Sales each month

Which graph should be used to summarize quartiles, medians, and outliers?

Boxplots should be used when you want to summarize data (the five number summary). Boxplots show the five number summary, skewness, outliers and spread.

This includes:

1. Minimum

2. Q1

3. Median

4. Q3

5. Maximum

   1. Or outliers

Anthony scored 76% on a midterm and was at the 68th percentile in a class of 125 students. Approximately how many students scored as well as or better than Anthony?

Percentile Rank

Look for:

• "68th percentile"

• "90th percentile"

• "What proportion is above/below?"

• "How many students scored better?"

68th percentile means 68% of students scored at or below Anthony.

• 100 - 68 = 32%. That means that 32% of students scored better than Anthony.

Find the position of the 75th percentile in a data set containing 40 observations.

Percentile Position

Look for:

• Percentile

• P75

• P20

• P90

• Position

Which interval contains the first quartile of a frequency distribution?

Looking for the quartile position.

Look for:

• Q1

• Q3

• First quartile

• Third quartile

• Frequency table

A survey has a relative frequency of 0.23 for Geography. There were 300 students surveyed. How many students chose Geography?

Frequency

Formula: Frequency = Relative Frequency x n (observations)

Look for:

• Relative frequency

• Frequency

• Total surveyed

English has a frequency of 60 and a relative frequency of 0.20. How many students were surveyed?

Missing Total from Relative Frequency

Look for:

• Frequency given

• Relative frequency given

• Total missing

Out of 40 students, 26 passed. What percentage passed?

Percentage

Look for:

• What percent?

• Percentage passed?

• Percentage failed?

Out of 50 taxi fares, 37 were less than $25. What proportion were less than $25?

Proportion

Look for:

• Proportion

• Relative frequency

The mean GPA of 14 students is 3.12. One student with GPA 2.41 leaves. Two students with GPAs 3.97 and 4.26 join. What is the new mean?

Mean Change / Missing Mean

Look for:

• Mean given

• Someone leaves

• Someone joins

• New mean needed

Five values have a mean of 20. Four values are 15, 18, 22, and 24. Find the missing value.

Missing Value from Mean

Mean Average x n (observations) - (sum of the total known values)

1. Find the total sum of all 8 observations by adding them all together (e.g. 15 + 18 + 22 + 24 = 79).

2. Multiply this number (79) by n (the total observations, even the one that we are not sure of). 79 × 5 = 395.

3. Add the total of the know numbers = 79.

4. Find the missing value by 395 - 79 = 316.

   1. Missing value = average x n - (sum of the whole values)

Look for:

• Mean known

• One observation missing

Section A has 123 students with an average score of 70.2. Section B has an average score of 62.8.

The combined average is 65.6. Find the number of students in Section B.

Weighted Mean

The average itself is never the weight. Whatever answers “how many” is usually the weight.

Look for:

• Two groups

• Two averages

• Overall average

• Number of students

A course is graded:

• Assignments = 20%

• Midterm = 30%

• Final = 50%

A student earns:

• 80% on assignments

• 70% on the midterm

Weighted grade

How to Read Timplots

A timeplot shows how something changes over time. Use for measurements, and not distributions.

The x-axis is always:

Time

Examples:

• Days

• Weeks

• Months

• Years

The y-axis is the measurement.

Examples:

• Temperature

• Stock Price

• Sales

• Population

Histogram

A histogram shows the distribution of quantitative (numerical) data.

• Histograms can tell you: Right skew, Left skew, Symmetric

Boxplots

Boxplots can be useful to summarize data (five number summary) related to:

1. Finding outliers

2. Comparing groups

3. Finding IQR

4. Finding skewness

5. Finding the median

They are NOT GOOD for:

• Seeing exact data and frequencies

• Showing trends over time

A dataset contains 250 observations.

Find the position of:

a) 25th percentile

b) 60th percentile

c) 90th percentile

Percentile Position

Use formula

Pk = k(n + 1)/100

• k: This represents the percentile number that you are trying to find. When something appears as a 25th, it can be left as 25, and does not need to be changed to 0.25.

• This it looking at understanding the percentile position.

A test has 100 students.

The 75th percentile score is 84.

Approximately how many students scored below 84?

75th percentile means 75% are below.

0.75(100) = 75 students

A histogram represents 80 exam scores.

12 students scored below 50.

What proportion scored below 50?

What percentage scored below 50?

Histogram proportion question.

A class of 30 students has an overall average of 72. 20 students have an average of 68. Find the average of the remaining 10 students.

Missing group mean

Question 19 is really asking you to compare two groups within a class. The entire class of 30 students has an average of 72, but one group of 20 students only averages 68, which is lower than the overall class average.

• Since those 20 students are pulling the average down, the remaining 10 students must have grades that are higher than 72 to bring the class average back up to 72 overall.

• Think of it like a tug-of-war: the larger group is below the average, so the smaller group has to be well above the average to balance everything out.

• When we calculate it, those remaining 10 students need an average of 80, which makes sense because 80 is higher than 72 and can offset the lower average of the first group.