Math Term 1

I. MEASURES OF CENTRAL TENDENCY

1. Mean

Definition: The mean is the average of a set of data.

Formula:

xˉ=∑xn\bar{x}=\frac{\sum x}{n}

Where:

• xˉ\bar{x} = mean

• ∑x\sum x = sum of all values

• nn = number of values

Quiz Bee Quick Recall

Q: What measure is also called the average?
A: Mean

Q: What do you do first when finding the mean?
A: Add all the values.

Q: What do you divide the sum by?
A: The number of values.

Example

Data:

6,8,10,126,8,10,126+8+10+12=366+8+10+12=36

There are 4 values.

36÷4=936\div4=9

Mean = 9

Memory Aid

MEAN = ADD THEN DIVIDE

2. Median

Definition: The median is the middle value when the data are arranged from least to greatest.

Example:

3,5,7,9,113,5,7,9,11

Median = 7

If there are two middle values, add them and divide by 2.

Example:

2,4,6,82,4,6,84+62=5\frac{4+6}{2}=5

Median = 5

Memory Aid

MEDIAN = MIDDLE

3. Mode

Definition: The mode is the value that occurs most frequently.

Example:

2,3,3,4,52,3,3,4,5

Mode = 3

Quick Facts

• One mode = unimodal

• Two modes = bimodal

• More than two modes = multimodal

• No value repeats = no mode

Memory Aid

MODE = MOST

Central Tendency Comparison

Quiz Bee Tip: If you see the words average, middle, or most frequent, immediately think mean, median, or mode.

II. MEASURES OF VARIABILITY

1. Variability

Definition: Variability describes how spread out the data values are.

A small variability means values are close together.

A large variability means values are more spread out.

2. Range

Formula:

Range=Highest Value−Lowest Value\text{Range}=\text{Highest Value}-\text{Lowest Value}

Example:

4,7,10,154,7,10,1515−4=1115-4=11

Range = 11

Memory Aid

RANGE = HIGH - LOW

3. Mean Deviation

Mean deviation tells how far the values are from the mean on average.

Formula:

MD=∑∣x−xˉ∣nMD=\frac{\sum|x-\bar{x}|}{n}

Steps

1. Find the mean.

2. Subtract the mean from each value.

3. Take the absolute value.

4. Add the deviations.

5. Divide by nn.

Example

2,4,62,4,6

Mean:

2+4+63=4\frac{2+4+6}{3}=4

Deviations:

∣2−4∣=2|2-4|=2∣4−4∣=0|4-4|=0∣6−4∣=2|6-4|=2MD=2+0+23=43≈1.33MD=\frac{2+0+2}{3}=\frac43\approx1.33

Mean deviation = 1.33

Memory Aid

MD = DISTANCE FROM MEAN

4. Sample Variance

Sample variance measures the average squared distance from the mean.

Formula:

s2=∑(x−xˉ)2n−1s^2=\frac{\sum(x-\bar{x})^2}{n-1}

Steps

1. Find the mean.

2. Subtract the mean from every value.

3. Square each difference.

4. Add the squared differences.

5. Divide by n−1n-1.

Important Quiz Bee Fact

Sample variance → divide by n−1n-1

Example:

2,4,62,4,6

Mean = 4

Squared deviations:

4,0,44,0,4

Sum:

88

Since n=3n=3:

n−1=2n-1=2s2=82=4s^2=\frac82=4

Sample variance = 4

Memory Aid

SAMPLE = ONE LESS

Sample → n−1n-1

5. Sample Standard Deviation

Standard deviation is the square root of variance.

Formula:

s=∑(x−xˉ)2n−1s=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}

If sample variance is 4:

s=4=2s=\sqrt4=2

Standard deviation = 2

Memory Aid

STANDARD DEVIATION = SQUARE ROOT OF VARIANCE

Variability Quick Reference

VERY IMPORTANT

Mean → nn
Mean deviation → nn
Sample variance → n−1n-1
Sample standard deviation → n−1n-1

III. ANALYZING AND INTERPRETING DATA

Important Terms

Pattern: A repeated or consistent arrangement in data.

Relationship: How two sets of values or variables are connected.

Difference: How values or groups compare.

Trend: The general direction of data, such as increasing or decreasing.

Outlier: An unusually high or low value compared with the rest of the data.

Example

Data:

10,11,12,13,5010,11,12,13,50

50 is unusually high.

It can pull the mean upward.

Quiz Bee Questions

Q: What is a value that is unusually high or low?
A: Outlier

Q: What measure is most affected by extreme values?
A: Mean

Q: What describes the general direction of data?
A: Trend

Memory Aid

OUTLIER = OUT OF LINE

IV. BASIC CONCEPTS OF PROBABILITY

1. Experiment

An experiment is an activity or process that produces a well-defined set of results or outcomes.

Examples:

• Tossing a coin

• Rolling a die

• Spinning a spinner

2. Outcome

An outcome is one possible result of an experiment.

Example:

For a coin:

HH

is one outcome.

TT

is another.

3. Sample Space

The sample space is the set of all possible outcomes.

Coin:

S={H,T}S=\{H,T\}

Die:

S={1,2,3,4,5,6}S=\{1,2,3,4,5,6\}

Your test specifically defines the sample space as the set of all possible outcomes of a random experiment.

4. Event

An event is a set of one or more outcomes.

Example:

Rolling an even number:

E={2,4,6}E=\{2,4,6\}

Outcome vs. Event

Outcome: One result.

Event: One or more results.

Memory Aid

O = ONE

E = EITHER ONE OR MORE

V. COUNTING OUTCOMES

1. Systematic Listing

List every possible outcome in an organized way.

Example:

Coin + die:

H1, H2, H3, H4, H5, H6
T1, T2, T3, T4, T5, T6

Total:

1212

This matches the die-and-coin type of problem in your test.

Best Used When:

There are only a small number of combinations.

2. Table

A table organizes choices into rows and columns.

Example:

2 shirts × 3 pants:

2×3=62\times3=6

Best Used When:

You want to clearly see every combination.

3. Tree Diagram

A tree diagram uses branches to show choices made in stages.

Example:

2 drinks and 3 snacks:

2×3=62\times3=6

Best Used When:

The experiment involves several stages or choices.

4. Fundamental Counting Principle

If one event can happen in aa ways and another can happen in bb ways:

a×ba\times b

total outcomes are possible.

Example:

3 shirts × 2 pants:

3×2=63\times2=6

Your test uses this exact type of problem.

Memory Aid

FCP = MULTIPLY THE CHOICES

VI. THEORETICAL AND EXPERIMENTAL PROBABILITY

Theoretical Probability

Based on all possible outcomes.

Formula

P(E)=favorable outcomestotal possible outcomesP(E)=\frac{\text{favorable outcomes}}{\text{total possible outcomes}}

Example:

Probability of rolling a 6:

P(6)=16P(6)=\frac16

Experimental Probability

Based on actual results from an experiment.

Formula

P(E)=number of times event occurstotal number of trialsP(E)=\frac{\text{number of times event occurs}}{\text{total number of trials}}

Your test explicitly identifies experimental probability as being based on the actual results of an experiment.

Example:

A die is rolled 60 times and lands on 6 ten times:

P(6)=1060=16P(6)=\frac{10}{60}=\frac16

This is also directly represented in your test.

Probability Facts to Memorize

0≤P(E)≤10\leq P(E)\leq1

0 = impossible

1 = certain

Between 0 and 1 = possible but not certain

Memory Aid

0 = NO

1 = SURE

Coin

Two possible outcomes:

H,TH,T

Probability of heads:

12\frac12

Probability of tails:

12\frac12

Die

Six possible outcomes:

1,2,3,4,5,61,2,3,4,5,6

Probability of one specific number:

16\frac16

Probability of an odd number:

36=12\frac36=\frac12

Cards

A standard deck has 52 cards.

There are 4 aces.

P(Ace)=452=113P(\text{Ace})=\frac4{52}=\frac1{13}

Spinner

If 1 out of 5 equal sections is red:

P(red)=15P(\text{red})=\frac15

VII. ALGEBRAIC EXPRESSIONS

Algebraic Expression

An algebraic expression contains numbers, variables, and mathematical operations.

Examples:

x+5x+53x−73x-72a+42a+4

Translation Key

Memory Aid

SUM = ADD

DIFFERENCE = SUBTRACT

PRODUCT = MULTIPLY

QUOTIENT = DIVIDE

Real-Life Modeling

Juan earns Php 50 per hour and pays a Php 100 fee.

For hh hours:

50h−10050h-100

Your test contains this exact type of algebraic modeling question.

Another example:

xx notebooks cost Php 25 each.

yy pens cost Php 10 each.

Total:

25x+10y25x+10y

If paying Php 500:

500−(25x+10y)500-(25x+10y)

This is also represented in your test.

VIII. ADDING AND SUBTRACTING MONOMIALS

Monomial

A monomial is an expression with one term.

Examples:

5x,3x2,7,−2ab5x,\quad 3x^2,\quad 7,\quad -2ab

Like Terms

Like terms have:

1. The same variables

2. The same exponents

Examples:

3x, 7x3x,\ 7x4a2, 9a24a^2,\ 9a^2

Rule

Add or subtract the coefficients and keep the variable part.

Example:

3x+5x=8x3x+5x=8x

Example:

9a2−4a2=5a29a^2-4a^2=5a^2

Unlike Terms

Examples:

3x, 4y3x,\ 4y5x2, 7x5x^2,\ 7x

They cannot be combined.

Memory Aid

LIKE TERMS = SAME LETTER, SAME POWER

IX. LAWS OF EXPONENTS

Vocabulary

In:

535^3

5 = base

3 = exponent

535^3 = power

The exponent tells how many times the base is multiplied by itself.

53=5×5×55^3=5\times5\times5

1. Product Rule

When multiplying powers with the same base, add exponents.

am⋅an=am+na^m\cdot a^n=a^{m+n}

Example:

x3⋅x4=x7x^3\cdot x^4=x^7

Memory Aid

MULTIPLY → ADD

2. Quotient Rule

When dividing powers with the same base, subtract exponents.

aman=am−n\frac{a^m}{a^n}=a^{m-n}

Example:

x8x3=x5\frac{x^8}{x^3}=x^5

Memory Aid

DIVIDE → SUBTRACT

3. Power of a Power

Multiply the exponents.

(am)n=amn(a^m)^n=a^{mn}

Example:

(x3)2=x6(x^3)^2=x^6

Memory Aid

POWER ON POWER → MULTIPLY

4. Power of a Product

Apply the exponent to every factor.

(ab)n=anbn(ab)^n=a^nb^n

Example:

(2x)3=8x3(2x)^3=8x^3

5. Power of a Quotient

Apply the exponent to the numerator and denominator.

(ab)n=anbn\left(\frac ab\right)^n=\frac{a^n}{b^n}

Example:

(x2)3=x38\left(\frac{x}{2}\right)^3=\frac{x^3}{8}

6. Zero Exponent

Any nonzero number raised to zero equals 1.

a0=1a^0=1

Example:

90=19^0=1

Memory Aid

ZERO POWER = ONE

7. Negative Exponent

A negative exponent means reciprocal.

a−n=1ana^{-n}=\frac1{a^n}

Example:

x−3=1x3x^{-3}=\frac1{x^3}

Memory Aid

NEGATIVE = FLIP

X. QUIZ BEE RAPID-FIRE ROUND

Try answering these without looking at the answers first.

Central Tendency

1. What is the average of a data set called?

2. What measure identifies the middle value?

3. What measure identifies the most frequent value?

4. What should you do before finding the median?

5. What happens when there are two middle values?

Variability

6. What is the highest value minus the lowest value?

7. What does variability describe?

8. What does mean deviation measure?

9. What do you divide by when finding mean deviation?

10. What do you divide by when finding sample variance?

11. What is the square root of variance?

12. What is another name for an unusually high or low value?

Probability

13. What is an experiment?

14. What is one possible result of an experiment?

15. What is the set of all possible outcomes?

16. What is a set of one or more outcomes?

17. What is the probability of an impossible event?

18. What is the probability of a certain event?

19. What probability is based on actual results?

20. What probability is based on possible outcomes?

Counting

21. What principle tells you to multiply the number of choices?

22. How many outcomes are there when rolling a die and tossing a coin?

23. What method uses branches to show choices?

24. What method uses rows and columns?

25. What method lists outcomes one by one?

Algebra

26. What does "product" mean?

27. What does "difference" mean?

28. What does "twice a number" mean?

29. What are like terms?

30. Can 3x3x and 4y4y be combined?

Exponents

31. What is the number being raised to a power called?

32. What tells how many times the base is used as a factor?

33. When multiplying same bases, what happens to the exponents?

34. When dividing same bases, what happens to the exponents?

35. What happens in a power of a power?

36. What is any nonzero number raised to zero?

37. What does a negative exponent indicate?

XI. ANSWERS TO RAPID-FIRE

1. Mean

2. Median

3. Mode

4. Arrange the data from least to greatest

5. Average the two middle values

6. Range

7. How spread out the data are

8. The average distance of values from the mean

9. nn

10. n−1n-1

11. Standard deviation

12. Outlier

13. An activity or process that produces possible results

14. Outcome

15. Sample space

16. Event

17. 0

18. 1

19. Experimental probability

20. Theoretical probability

21. Fundamental Counting Principle

22. 12

23. Tree diagram

24. Table

25. Systematic listing

26. Multiplication

27. Subtraction

28. 2x2x

29. Terms with the same variables and exponents

30. No

31. Base

32. Exponent

33. Add them

34. Subtract them

35. Multiply them

36. 1

37. A reciprocal

XII. MUST-MEMORIZE QUIZ BEE FACTS

Statistics

Mean = Average

Median = Middle

Mode = Most

Range = Highest − Lowest

Outlier = Unusually high or low value

Sample variance = divide by n−1n-1

Sample standard deviation = square root of variance

Probability

Experiment → produces results

Outcome → one result

Sample Space → all possible results

Event → one or more outcomes

Theoretical → possible outcomes

Experimental → actual results

Impossible = 0

Certain = 1

Probability range = 0 to 1

Counting

Table → rows and columns

Tree → branches

Systematic listing → list everything

FCP → multiply choices

Algebra

Sum → +

Difference → −

Product → ×

Quotient → ÷

Twice → ×2

Three times → ×3

Like terms → same variables + same exponents

Exponents

Multiply same bases → ADD exponents

Divide same bases → SUBTRACT exponents

Power of a power → MULTIPLY exponents

Zero exponent → 1

Negative exponent → reciprocal

XIII. STRATEGIC QUIZ BEE TIPS

Tip 1: Look for trigger words

If you hear:

• average → Mean

• middle → Median

• most often → Mode

• highest and lowest → Range

• actual experiment → Experimental probability

• possible outcomes → Theoretical probability

• all possible outcomes → Sample space

• one possible result → Outcome

• multiply choices → Fundamental Counting Principle

• same terms → Like terms

• same base + multiplication → Add exponents

Tip 2: For probability, use F/T

Think:

F/T = Favorable ÷ Total

Tip 3: For counting, ask "AND or OR?"

AND → usually multiply

Example:

3 shirts and 2 pants:

3×2=63\times2=6

Tip 4: For exponents, remember MADD

Multiply → ADD exponents

Then:

Divide → Subtract

Tip 5: Watch nn vs. n−1n-1

This is a common trap.

Sample = subtract 1

n−1n-1

Tip 6: In a tie, eliminate wrong answers quickly

For multiple-choice questions, first identify the type of question. You can often eliminate options that do not match the required formula, unit, or concept before calculating.

Final Super-Mnemonic

"Mean Middle Most, High-Low Range, Sample One Less, F over T, Choices Multiply, Like Terms Match, Multiply Add, Divide Subtract."