Understanding Mean, Median, and Mode Ratios

The question asks us to find the ratio of Median to Mode (B : C) given the ratio of Mean to Median (A : B). We are given that A, B, and C represent the Mean, Median, and Mode of a data set, respectively. The ratio A : B is given as 9 : 8.

Relating Mean, Median, and Mode

For a moderately skewed distribution, there is an empirical relationship between the Mean, Median, and Mode. This relationship is often stated as:

$$ \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} $$

Using the given notations A, B, and C for Mean, Median, and Mode:

$$ C \approx 3B - 2A $$

Calculating the Ratio B : C

We are given the ratio A : B = 9 : 8. This means that $\frac{A}{B} = \frac{9}{8}$.

We can express A and B in terms of a common constant, say 'k'. Let:

• A = 9k
• B = 8k

Here, 'k' is a non-zero constant.

Now, substitute these values into the empirical relationship \(C = 3B - 2A\):

$$ C = 3(8k) - 2(9k) $$

Perform the multiplications:

$$ C = 24k - 18k $$

Subtract the terms:

$$ C = 6k $$

So, we have B = 8k and C = 6k.

We need to find the ratio B : C. This is $\frac{B}{C}$.

$$ \frac{B}{C} = \frac{8k}{6k} $$

Cancel out the common factor 'k' (since k is non-zero):

$$ \frac{B}{C} = \frac{8}{6} $$

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$$ \frac{B}{C} = \frac{8 \div 2}{6 \div 2} = \frac{4}{3} $$

Thus, the ratio B : C is 4 : 3.

Comparing with Options

Let's compare our calculated ratio B : C = 4 : 3 with the given options:

• Option 1: 5 : 4
• Option 2: 8 : 9
• Option 3: 7 : 6
• Option 4: 4 : 3

Our calculated ratio matches Option 4.

Summary of Calculation Steps

The ratio of B (Median) to C (Mode) is 4 : 3.

Revision Table: Measures of Central Tendency

Additional Information: Empirical Relationship

The empirical relationship Mode \(\approx\) 3 Median - 2 Mean is an approximate relationship that holds true for unimodal distributions that are moderately skewed. It's a useful rule of thumb when the exact relationship is unknown or difficult to calculate directly from the data.

Key points about the relationship:

• It connects the three main measures of central tendency.
• It's based on observations of many different datasets, not a strict mathematical proof that applies to all distributions.
• For a perfectly symmetrical distribution (like a normal distribution), Mean = Median = Mode. In this case, the formula holds true: Mean = 3(Mean) - 2(Mean).
• The formula is particularly helpful in situations where one of the measures is difficult to find directly, but the other two are known.