The problem asks us to find the value of '$b$' given a dataset $\{-5, 1, a, 5, b\}$, its ascending order property, and its mean and median values.
Dataset Properties Analysis
- The dataset elements are $\{-5, 1, a, 5, b\}$.
- The elements are in ascending order, which means: $-5 \le 1 \le a \le 5 \le b$
- From the ascending order, we deduce the constraints: $a \ge 1$, $a \le 5$, and $b \ge 5$.
Median Calculation
The dataset has 5 elements. The median is the middle element in the sorted dataset. In this case, the middle element is '$a$'.
Given that the median is 3, we can set '$a$' equal to 3:
$a = 3$This value of $a=3$ satisfies the condition $1 \le a \le 5$, derived from the ascending order property.
Mean Calculation
The mean of a dataset is the sum of its elements divided by the number of elements.
The formula for the mean is:
$ \text{Mean} = \frac{\sum \text{elements}}{\text{Number of elements}} $Substitute the given values:
$ \text{Mean} = \frac{-5 + 1 + a + 5 + b}{5} $We are given that the mean is 3. Substitute $a=3$ into the equation:
$ 3 = \frac{-5 + 1 + 3 + 5 + b}{5} $Simplify the numerator:
$ 3 = \frac{4 + b}{5} $To solve for '$b$', multiply both sides by 5:
$ 3 \times 5 = 4 + b $ $ 15 = 4 + b $Isolate '$b$' by subtracting 4 from both sides:
$ b = 15 - 4 $ $ b = 11 $Final Value Verification
The calculated value is $b=11$. Let's check if this fits all conditions:
- Dataset: $\{-5, 1, 3, 5, 11\}$.
- Ascending order: $-5 \le 1 \le 3 \le 5 \le 11$. This is correct.
- Median: The middle element is 3. This is correct.
- Mean: $ \frac{-5 + 1 + 3 + 5 + 11}{5} = \frac{15}{5} = 3 $. This is correct.
The value $b=11$ satisfies all the conditions of the problem.
Important Questions from Mean Median Mode
- The median of the following set of numbers: 154, 130, 144, 137, 156, 146, 138, 149, 160, 138 is:
- Let $X$ be a continuous random variable whose cumulative distribution function (CDF)
$F_X(x) = \begin{cases} 0 & x < t \\ \frac{x-t}{4-t} & t \le x \le 4 \\ 1 & x \ge 4 \end{cases}$
If the median of $X$ is $3$, then what is the value of $t$? - The sample average of 50 data points is 40. The updated sample average after including a new data point taking the value of 142 is ________.
The above frequency chart shows the frequency distribution of marks obtained by a set of students in an exam. From the data presented above, which one of the following is CORRECT?
- The frequency distribution of beak sizes of a bird species is symmetric but not normally distributed. If the mean value of beak size is 6 mm, standard deviation is 25 mm and kurtosis is 10, then the median is ______ mm.
