Given Data:

• $x_i$: 0, 1, 2, 3, 4, 5
• $f_i$: 3, 3, $p$, 7, $p - 1$, 4
• Mean ($\bar{x}$) is given as 26.

We need to:

1. Find the value of $p$.
2. Find the missing observation values.
3. Find the mode of the data.

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Step 1: Formula for Mean

The formula for the mean of a frequency distribution is: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ Substituting the given values: $26 = \frac{\sum f_i x_i}{\sum f_i}$

(a) Calculate $\sum f_i x_i$

Using the given data: $\sum f_i x_i = (3 \times 0) + (3 \times 1) + (p \times 2) + (7 \times 3) + [(p-1) \times 4] + (4 \times 5)$ $\sum f_i x_i = 0 + 3 + 2p + 21 + 4p - 4 + 20$ $\sum f_i x_i = 40 + 6p$

(b) Calculate $\sum f_i$

$\sum f_i = 3 + 3 + p + 7 + (p-1) + 4$ $\sum f_i = 16 + 2p$

(c) Substituting into the mean formula

$26 = \frac{40 + 6p}{16 + 2p}$ Cross-multiply: $26(16 + 2p) = 40 + 6p$ $416 + 52p = 40 + 6p$ Simplify: $52p - 6p = 40 - 416$ $46p = -376$ $p = -\frac{376}{46} = -8$

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Step 2: Find the Value of Observations

Substituting $p = -8$:

• $f_2 = p = -8$
• $f_4 = p - 1 = -9$

Step 3: Mode

Mode is the value of $x_i$ corresponding to the highest $f_i$.
With $f_i$: $f_i = 3, 3, -8, 7, -9, 4$ Clearly, $x_i = 3$ (corresponding to $f_3 = 7$) is the mode.

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Final Results:

1. $p = -8$.
2. Observations: $f_2 = -8$, $f_4 = -9$.
3. Mode = $x_i = 3$.

Would you like further clarification or details? 😊

5 Related Questions:

1. How do you calculate the median for grouped data?
2. What happens when a frequency value becomes negative in a distribution?
3. How can the mode be determined graphically?
4. What are the implications of a negative mean?
5. How does the mean affect the choice of mode?

Tip:

In frequency distributions, always ensure that all $f_i$ are valid (non-negative) to maintain logical consistency in data interpretation.