Basics of ML & AL Week 2 Challenge
- 1. Calculate all 4 business moments using pen and paper for the below data set.
|
x |
x-mean |
(x-mean)^2 |
z value for x |
|
1 |
-2 |
4 |
-1.264911064 |
|
2 |
-1 |
1 |
-0.632455532 |
|
3 |
0 |
0 |
0 |
|
4 |
1 |
1 |
0.632455532 |
|
5 |
2 |
4 |
1.264911064 |
|
mean |
3 |
||
|
std |
1.581139 |
|
y |
y-mean |
(y-mean)^2 |
z value for y |
|
100 |
-70 |
4900 |
-1.455650686 |
|
150 |
-20 |
400 |
-0.415900196 |
|
175 |
5 |
25 |
0.103975049 |
|
200 |
30 |
900 |
0.623850294 |
|
225 |
55 |
3025 |
1.143725539 |
|
mean |
170 |
||
|
std |
48.08846 |
- 2. What is the significance of expected value when simple mean (Sum of all observations/number of observations) is already in place
- Ø The expected value and the simple mean are closely related concepts. The expected value is essentially a generalization of the mean in the context of probability and statistics. While the simple mean is a measure of central tendency based on observed data, the expected value considers the probabilities associated with different outcomes.
- Ø For a discrete random variable, the expected value is calculated by summing the product of each possible value and its probability. For a continuous random variable, it involves integrating the product of the variable and its probability density function.
- Ø So, even though the simple mean may be appropriate for describing the center of a set of observed data, the expected value is particularly useful in situations where you're dealing with uncertain or random events. It's a way to incorporate probabilities into the measure of central tendency, providing a more comprehensive understanding, especially in the realm of probability theory and statistics.
- 3. Is skewness in the curve considered to be bad in the analysis?
- 1. Interpretation of Skewness:
- Positive Skewness:
- If your data has a positive skewness, it means that the right tail is longer. This could imply that there are some high values (outliers) in the data. In some cases, positive skewness may be desirable, especially in financial contexts where you might want to capture the potential for large positive returns.
- Negative Skewness:
- Negative skewness indicates a longer left tail. This might suggest that there are more lower values in the data. Negative skewness is not necessarily bad; it depends on the context. For example, in finance, it might indicate a greater risk of large negative returns.
- 2. Impact on Analysis:
- Skewness can affect the assumptions of some statistical tests. For instance, normality is assumed in many parametric statistical tests, and skewness can violate this assumption. In such cases, transformations or alternative statistical tests may be considered.
- 3. Consideration of Other Measures:
- Skewness should be considered alongside other measures of central tendency and dispersion. Sometimes a distribution may have skewness, but if the mean and median are close, the impact of skewness might be minimal.
- 4. Evaluate the probabilities for continuous normal distribution with given mean = 680 and standard deviation = 31
- Ø X is the random variable from the normal distribution,
- Ø μ is the mean of the distribution,
- Ø σ is the standard deviation of the distribution,
- Ø Z is the standard score (z-score).
- a) P(X<711)
- Z=31(711−680)
- b) P(X>740)
- Z=31(740−680)
- c) P(600<X<720)
- 2=(720−680)31Z2=31(720−680)
- d) P(X=720)
- (for a continuous distribution, the probability of a single point is technically zero).
- 5. Explain the curve on the right side.
- Peak of the Curve:
- The highest point of the curve corresponds to the mean of the distribution. In your case, with a mean (μ) of 680, the peak is at x=680.
- Probability Density:
- The height of the curve at any given point represents the probability density of the random variable at that point. While the probability of a specific point is technically zero for a continuous distribution, the area under the curve within a range of values represents the probability of the variable falling within that range.
- Right Tail:
- As you move to the right from the mean, the curve gradually decreases. This indicates that values on the right side of the mean are less probable. The tail extends indefinitely, theoretically covering all possible real numbers.
- Z-Score on the Right Side:
- As you move to the right along the x-axis, you are moving into positive z-scores. A positive z-score indicates how many standard deviations a data point is from the mean. The farther to the right, the larger the z-score, suggesting values that are increasingly above the mean.