Question

delve into the world of film criticism by analyzing the meta - score variable to calculate the range, variance, and standard deviation. these statistical measures offer a deeper understanding of how movie ratings vary, shedding light on the diversity and depth of critical opinions. in the realm of film criticism, understanding the range, variance, and standard deviation of meta - scores can be enlightening for movie enthusiasts and filmmakers alike. the range illustrates the full spectrum of critical reception, from acclaimed masterpieces to panned flops. the variance quantifies the extent of divergence in critic opinions, and the standard deviation provides a measure of how far, on average, scores deviate from the mean meta - score. use the following subset of the imdb dataset to answer the questions: 1. what is the range? what does that tell us? the range is 38, indicating the span between the highest and lowest ratings. (round to one decimal place as needed.) 2. what is the standard deviation? the standard deviation is (round to three decimal places as needed.) 3. describe the spread and what that tells us about how films are appraised by critics. the standard deviation signifies that the of individual ratings from the rating is (round to three decimal places as needed.)

Explanation:

Step1: Recall range formula

Range = Maximum - Minimum

Step2: Identify maximum and minimum

From the data set (not shown completely here but we know the range is already given as 38 which means we assume the max - min is already calculated correctly), we understand that the range shows the difference between the highest and lowest ratings. It gives a simple measure of the spread of the data values.

Step3: Recall standard - deviation formula

The formula for the sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\), where \(x_{i}\) are the individual data points, \(\bar{x}\) is the sample mean and \(n\) is the sample size.

Step4: Calculate standard deviation (assuming data values \(x_1,x_2,\cdots,x_n\))

First, calculate the mean \(\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}\). Then for each data - point \(x_{i}\), find the deviation \((x_{i}-\bar{x})\), square it \((x_{i}-\bar{x})^{2}\), sum these squared deviations \(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\), divide by \(n - 1\) and then take the square - root.
Let's assume the data set is \(x_1,x_2,\cdots,x_n\). The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\).
\(\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}=(x_1 - \bar{x})^{2}+(x_2-\bar{x})^{2}+\cdots+(x_n-\bar{x})^{2}\)
\(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\)

Step5: Interpret standard deviation

The standard deviation measures how far, on average, each value in the data set is from the mean. A small standard deviation indicates that the data points tend to be close to the mean, while a large standard deviation indicates that the data points are spread out over a wider range of values.

Answer:

1. The range is 38. It tells us the span between the highest and lowest ratings of movies by critics.
2. (Calculation of standard deviation requires the full data set. Let's assume the data set has \(n\) values \(x_1,x_2,\cdots,x_n\)). First, find the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Then calculate \(\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\)[LLM SSE On Failure]