Quantitative Analysis for Business

How do you find the limits for the class in statistics?

1. Determine the interval range: Decide the range of data values to be grouped.

2. Choose a class width: This is the difference between the upper and lower limits of a class, calculated from the data.

3. Set the lower limit of the first class: This is usually the smallest data value or a rounded value below it. Since the class mark is the exact center of the class, the boundaries are located half a class width (10 / 2 = 5) away from the mark.

4. Calculate the upper limit: Add the class width to the lower limit to find the upper limit. For example, if the lower limit is 5 and the class width is 10, the upper limit would be 5 + 10 = 15.

How do you calculate the midpoint?

The midpoint is calculated by finding the average of the upper and lower limits of a class interval. You add the lower limit and upper limit together, then divide by 2.

Midpoint = (Lower Limit + Upper Limit) / 2

How do you calculate relative frequency?

The Core Formula

\text{Relative Frequency} = \frac{f}{n}

Where:

• f = The frequency of a specific class (how many times that value appears).

• n = The total number of observations (the sum of all frequencies).

What is the 2 to the k rule?

The 2^k rule is a quick "rule of thumb" used to determine the ideal number of classes (or bins) to use when you are constructing a frequency distribution.

It helps you avoid the two biggest mistakes in data visualisation:

1. Having too many classes (making the data look scattered and noisy).

2. Having too few classes (clumping everything together and hiding the patterns).

To find the number of classes (k), you choose the smallest whole number k such that 2 to the power of k is greater than or equal to the total number of observations (n).

2^k \ge n

Reference table for the 2 to the k rule

Number of Observations (n)	Suggested Classes (k)	2k Value
10 – 30	5	32
31 – 60	6	64
61 – 120	7	128
121 – 250	8	256
251 – 500	9	512
501 – 1,000	10	1,024

Compute the variance and standard deviation of {X, Y, Z}.

1. Mean: Find the average.

2. Deviations: Subtract the mean from every number.

3. Square: Square each of those results.

4. Sum: Add the squares together.

5. Variance (s): Divide the sum by (n-1).

6. Std Dev (s): Take the square root of the variance.

When and how do you use CV?

• When: Use it to compare two datasets with different means or units.

  • How: CV = Standard Deviation / Mean x 100

  • Interpretation: Higher % = more relative variability (riskier).

How do you turn a raw value into a percentage?

Z-score: Z = Value - Mean / Std Dev

2. Table: Look up the Z-score to find the Area to the Left.

3. Logic: If question says "Less than": Use the table value.

If question says "More than": 1 - Table Value

4. Percent: Multiply by 100.

How do you find a score (X) when given a percentage?

1. Find Area to Left: If given "top 5%", the area to the left is 0.95.

2. Find Z: Look in the middle of the Z-table for the area and find the matching Z-score (e.g., 0.95 to Z = 1.645).

3. Find X: Use the formula X = Mean + (Z x Std Dev)

How do you find variance from a P(X) table?

. Create a "Mean Column" ($X \cdot P(X)$)

Multiply each value by its probability and add them all up. This is your Mean (mu).

• Example: If X is 2 and P(X) is 0.1, the value is 0.2

2. Create an "X^2" Column

Square every X value before looking at the probabilities.

• Example: If X is 3, X² is 9.

3. Create an "X^2 x P(X)" Column

Multiply your new squared numbers by the original probabilities and add them up.

• Example: If X² is 9 and P(X) is 0.1, the value is 0.9.

4. The Final Subtraction (The "Big Formula")

Subtract the Mean squared from the sum you got in Step 3.

\text{Variance} = (\text{Sum of } X^2 \cdot P(X)) - (\mu)^2

What is the formula for spread in a Uniform (rectangle) distribution?

Variance: (Max - Min)² / 12

Standard Deviation: sqrt{Variance}

• Note: Always divide by 12—it's a mathematical constant for this distribution type!

Empirical Rule (68-95-99.7)

The Empirical Rule states that for a normal distribution:

• 68% of the data falls within +1 standard deviation.

• 95% of the data falls within +2 standard deviations.

• 99.7% of data falls within +3 standard deviations.

Multiply the Standard Deviation by 2 for 95%

2. The Calculation Steps

To find the two amounts that trap the middle 95%, use these two formulas:

• Lower Amount = Mean - (2 x Standard Deviation)

• Upper Amount = Mean + (2 x Standard Deviation)

The 2^k Rule vs. Professor's Rules

The Rule: Find the lowest k where 2^k > n (for n=25, 2^5=32, so k=5).

• The Exception: If your key says 4, they are likely using a simpler "square root" rule or adjusting for a small sample size to avoid empty classes.

• Action: Always default to the specific number of classes your textbook or professor uses in their examples!

New notes

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TheMean(\bar{x})

• Logic: The "Average."

• How to do it: Sum all the numbers in the series and divide by the count (n).

TheVariance(s^2):

• Logic: The average of the squared distances from the mean.

• How to do it: 1. Subtract the Mean from every number (to see the "gap").

  2. Square those gaps (so they are all positive).

  3. Add them all up.

  4. Divide by (n-1)

TheStandardDeviation(s):

• Logic: The "Standard Gap." It puts the variance back into the original units (e.g., dollars instead of dollars-squared).

• How to do it: Take the Square Root of the Variance.

Comparing Variability with Different Units

• The Tool to use: Coefficient of Variation (CV).

• The Formula:

  CV=\left(\frac{s}{\bar{x}}\right)\times100

• Why use it? It expresses the standard deviation as a percentage of the mean. This "washes away" the units (Gallons vs. Dollars).

• How to interpret: The series with the higher CV percentage has more "relative variability." It is the more "unpredictable" series relative to its size.