Math Problem Statement

Based on the graph of this normal distribution, what are the mean, median, mode, and standard deviation?

Solution

The image presents a bell curve that represents a normal distribution, with labeled values along the horizontal axis ranging from 22 to 70. Based on the symmetry of the normal distribution, we can determine key statistics:

  • The mean, median, and mode are all located at the center of the distribution because, for a perfectly normal distribution, these three values are equal.

From the graph:

  1. The center point of the curve aligns with 46, which is the:

    • Mean (a) = 46
    • Median (b) = 46
    • Mode (c) = 46
  2. The standard deviation can be estimated based on the spacing of the axis. Typically, for a normal distribution, one standard deviation away from the mean will include about 68% of the data. Here, if we observe the distance from the mean (46) to one point of inflection (either 38 or 54), it is roughly 8 units.

Thus, the answers are:

  • a. The mean is 46.
  • b. The median is 46.
  • c. The mode is 46.
  • d. The standard deviation is approximately 8.

Would you like more details or have any further questions?

Here are 5 related questions to expand on this:

  1. How do we calculate the variance from the standard deviation?
  2. What are the key properties of a normal distribution?
  3. How do we interpret z-scores in a normal distribution?
  4. What percentage of data falls within 2 standard deviations from the mean in a normal distribution?
  5. How do you compute probabilities using the standard normal distribution?

Tip: In a normal distribution, 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations from the mean. This is known as the Empirical Rule.

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Math Problem Analysis

Mathematical Concepts

Statistics
Normal Distribution
Measures of Central Tendency
Standard Deviation

Formulas

Mean = Sum of all values / Number of values
Standard Deviation = sqrt[Sum of (xi - x̄)^2 / N]

Theorems

Empirical Rule
68-95-99.7 Rule

Suitable Grade Level

Grades 9-12