If Each Observation Of The Data Is Decreased By 15; Then The Mean:a) Remains Same B) Is Decreased By 15c) Is Increased By 15 D) Is Multiplied By 15​

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Introduction

In statistics, the mean is a crucial measure of central tendency that represents the average value of a dataset. When working with data, it's essential to understand how changes in individual observations affect the overall mean. In this article, we'll explore the effect of decreasing each observation in a dataset by a fixed amount, specifically 15, on the mean.

What is the Mean?

The mean, also known as the arithmetic mean, is calculated by summing up all the observations in a dataset and dividing by the number of observations. Mathematically, it can be represented as:

μ = (Σx) / n

where μ is the mean, x represents each observation, Σ denotes the sum of all observations, and n is the number of observations.

Decreasing Observations by 15

Now, let's consider what happens when each observation in the dataset is decreased by 15. This means that each value in the dataset is reduced by 15. To understand the impact on the mean, we need to analyze the effect on the sum of observations and the number of observations.

Effect on the Sum of Observations

When each observation is decreased by 15, the sum of all observations is also reduced by 15 times the number of observations. Mathematically, this can be represented as:

Σ(x - 15) = Σx - 15n

where Σ(x - 15) is the sum of observations decreased by 15, Σx is the original sum of observations, and n is the number of observations.

Effect on the Mean

Now, let's substitute the new sum of observations into the formula for the mean:

μ = (Σ(x - 15)) / n

Substituting the expression for Σ(x - 15) from the previous step, we get:

μ = (Σx - 15n) / n

Simplifying the expression, we get:

μ = (Σx / n) - 15

This shows that the new mean, μ, is equal to the original mean, Σx / n, minus 15.

Conclusion

In conclusion, when each observation in a dataset is decreased by 15, the mean is also decreased by 15. This is because the sum of observations is reduced by 15 times the number of observations, which in turn reduces the mean by 15.

Key Takeaways

  • Decreasing each observation in a dataset by a fixed amount affects the mean.
  • The mean is decreased by the same amount as the decrease in each observation.
  • The effect on the mean is proportional to the number of observations.

Example

Suppose we have a dataset with the following observations: 10, 20, 30, 40, 50. The mean of this dataset is:

μ = (10 + 20 + 30 + 40 + 50) / 5 = 25

If we decrease each observation by 15, the new dataset becomes: -5, 5, 15, 25, 35. The new mean is:

μ = (-5 + 5 + 15 + 25 + 35) / 5 = 10

As we can see, the mean has decreased by 15, which is the same amount as the decrease in each observation.

Real-World Applications

Understanding the impact of decreasing observations on the mean has practical applications in various fields, such as:

  • Finance: When a company decreases its prices by a fixed amount, the mean price of its products or services also decreases.
  • Economics: When a country decreases its interest rates by a fixed amount, the mean interest rate of its economy also decreases.
  • Statistics: When a dataset is adjusted for outliers or errors, the mean may be affected, and understanding this impact is crucial for accurate analysis.

Conclusion

Q: What happens to the mean when each observation is increased by 15?

A: When each observation in a dataset is increased by 15, the mean is also increased by 15. This is because the sum of observations is increased by 15 times the number of observations, which in turn increases the mean by 15.

Q: Can the mean be increased by more than 15 if each observation is increased by a different amount?

A: Yes, the mean can be increased by more than 15 if each observation is increased by a different amount. For example, if each observation is increased by 20, the mean will be increased by 20, not just 15.

Q: What happens to the mean when each observation is decreased by a different amount?

A: When each observation in a dataset is decreased by a different amount, the mean will be decreased by the average of the decreases. For example, if each observation is decreased by 10, 15, and 20, the mean will be decreased by (10 + 15 + 20) / 3 = 15.

Q: Can the mean be decreased by more than 15 if each observation is decreased by a different amount?

A: Yes, the mean can be decreased by more than 15 if each observation is decreased by a different amount. For example, if each observation is decreased by 20, 25, and 30, the mean will be decreased by (20 + 25 + 30) / 3 = 25.

Q: What happens to the mean when a constant is added to each observation?

A: When a constant is added to each observation, the mean is also increased by the same constant. For example, if a constant of 10 is added to each observation, the mean will be increased by 10.

Q: Can the mean be increased by more than 10 if a constant is added to each observation?

A: Yes, the mean can be increased by more than 10 if a constant is added to each observation. For example, if a constant of 20 is added to each observation, the mean will be increased by 20.

Q: What happens to the mean when a constant is subtracted from each observation?

A: When a constant is subtracted from each observation, the mean is also decreased by the same constant. For example, if a constant of 10 is subtracted from each observation, the mean will be decreased by 10.

Q: Can the mean be decreased by more than 10 if a constant is subtracted from each observation?

A: Yes, the mean can be decreased by more than 10 if a constant is subtracted from each observation. For example, if a constant of 20 is subtracted from each observation, the mean will be decreased by 20.

Q: How does the mean change when a new observation is added to the dataset?

A: When a new observation is added to the dataset, the mean will change depending on the value of the new observation. If the new observation is greater than the previous mean, the mean will increase. If the new observation is less than the previous mean, the mean will decrease.

Q: Can the mean be affected by the order of the observations in the dataset?

A: No, the mean is not affected by the order of the observations in the dataset. The mean is calculated by summing up all the observations and dividing by the number of observations, regardless of the order in which they are listed.

Q: How does the mean change when a dataset is combined with another dataset?

A: When two datasets are combined, the mean of the combined dataset is the weighted average of the means of the two individual datasets. The weights are the number of observations in each dataset.

Q: Can the mean be affected by the presence of outliers in the dataset?

A: Yes, the mean can be affected by the presence of outliers in the dataset. Outliers are observations that are significantly different from the rest of the data. If an outlier is present in the dataset, it can pull the mean in the direction of the outlier, resulting in a mean that is not representative of the data.