If the ratio of mode and median is 7 : 4, then the ratio of mean and mode is:Select an answerA7 : 11B2 : 3C5 : 14D8 : 9

1. Break Down the Problem

• We are given the ratio of mode to median, which is $\frac{\text{Mode}}{\text{Median}} = \frac{7}{4}$.
• We need to find the ratio of mean to mode.

2. Relevant Concepts

• In statistics, the relationship between mode, median, and mean can vary based on the distribution. However, for many distributions, especially in a normal distribution, the following relationships hold: $\text{Mean} \approx \text{Median} \approx \text{Mode}$
• However, we can use a more general relationship between these measures. When the ratio of mode and median is known, we can derive the ratio of mean and mode.

3. Analysis and Detail

• Let’s denote:

  • Mode = $M$
  • Median = $Q$
  • Mean = $A$

• From the given ratio: $\frac{M}{Q} = \frac{7}{4} \Rightarrow M = \frac{7}{4}Q$

• For a symmetrical distribution such as a normal distribution, generally, we often find: $A \approx M + \frac{M - Q}{3}$

• In order to find the ratio $\frac{A}{M}$:

  • Substitute $Q = \frac{4}{7}M$ into the mean’s approximation: $A \approx M + \frac{M - \frac{4}{7}M}{3} = M + \frac{M(\frac{3}{7})}{3} = M + \frac{1}{7}M = \frac{8}{7}M$

• Thus, the ratio of mean to mode is: $\frac{A}{M} = \frac{\frac{8}{7}M}{M} = \frac{8}{7}$

4. Verify and Summarize

• The derived ratio of the mean to mode simplifies to $\frac{8}{7}$.
• We need to find which of the provided options can match this ratio. None of the given options represent $8:7$ precisely, however, the closest matching available ratios can be explored.

Final Answer

Based on the analysis above, the ratio of mean to mode is closest to $8:7$. Therefore, the correct option cannot be directly identified from the given choices since none match $8 : 7$.