Students in a large accounting class were asked to rate the...

Students in a large accounting class were asked to rate the course by assigning a score of $1,2,3,4$, or 5 to the course. A higher score indicates that the students received greater value from the course. The accompanying table shows proportions of students rating the course in each category. $$ \begin{array}{lccccc} \hline \text { Rating } & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Proportion } & 0.07 & 0.19 & 0.28 & 0.30 & 0.16 \\ \hline \end{array} $$ Find the mean and standard deviation of the ratings.

Students in a large accounting class were asked to rate the course by assigning a score of $1,2,3,4$, or 5 to the course. A higher score indicates that the students received greater value from the course. The accompanying table shows proportions of students rating the course in each category.
$$
\begin{array}{lccccc}
\hline \text { Rating } & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Proportion } & 0.07 & 0.19 & 0.28 & 0.30 & 0.16 \\
\hline
\end{array}
$$
Find the mean and standard deviation of the ratings.

Key Concepts

Standard Deviation

The standard deviation is a measure of the dispersion or spread of a distribution around its mean. It is obtained by taking the square root of the variance, which is the weighted average of the squared differences between each value and the mean. This measure indicates how much the individual ratings deviate from the mean rating.

Mean

The mean, or expected value, is a measure of the central tendency of a random variable. It is calculated by taking the weighted average of all possible values, using their corresponding probabilities as weights. This provides a single value that summarizes the overall rating.

Discrete Random Variable

A discrete random variable is one that takes on values from a countable set, such as integer values. In the context of ratings, each possible rating is a distinct value and the probabilities associated with these ratings form a discrete distribution.

Probability Distribution

A probability distribution describes all the possible values and probabilities that a discrete random variable can assume. It provides the framework for understanding the likelihood of different outcomes, such as ratings in a survey where each rating has an associated probability.

Transcript

0:00 Welcome to numerate.

00:02 In the current problem, we are given the variable x, which is the ratings for course, right? so, x is the ratings.

00:17 So, we have the ratings.

00:20 One, two, three, four, five.

00:28 And we have the probabilities correspondingly.

00:33 Given that 0 .07, 0 .19, 0 .28, 0 .3028, 0 .30 .16.

00:47 And as i always suggest, check if this sum is 1 or not, if not, don't proceed with the sum.

00:53 Just tell that it's not a correct probability distribution, hence computing expectation and variance will be of no use.

01:00 Now, for expectation, we know the formula is whatever is inside e, okay, that into probability, correct? that means what? what is inside? so, first thing is 1 into 0 .07, correct? plus 2 into 0 .19 plus 3 into 0 .28, plus 4 into 0 .3 plus 5 into 0 .16.

01:35 So, whatever answer you come up, that will be the expectation.

01:39 So, what is this? this column is x into px.

01:43 So what will be this value? i have already calculated and kept 0 .07, 0 .38, 0 .84, 1 .2, and 0 .8, and finally, this sums at 3 .29.

02:02 So, it is 3 .29.

02:05 Okay.

02:07 Now, the next thing that we have to find is the standard division of x, which is nothing but square root of variance.

02:15 Now, what is variance? variance is expectation of x minus x squared.

02:27 That means, whatever inside e, what is this, this expression? so, x minus x x.

02:37 Square into px.

02:40 What is that? x minus how much is ex? ex is 3 .29 correct? so, it is 3 .29 square into px...

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