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I have a set of data which is the result of a set of questionnaires for different classes.

  • The questionnaires are identical (but they refer to different subjects)
  • Each class has different number of participants.

The dataset I have has for each class and question:

  • Mean
  • std
  • no. of participants

Regarding the no of participants a few assumptions.

  • not all students in a class answer the questionnaire
  • Also, for simplicity sake assume that the number of students that participate $n_{i,j}$ is much smaller than the total students in each class
  • the number of students that participate is greater than 30 (i.e. $n_{i,j}>30$)

e.g.

Class Question mean std no.Participants
$C_1$ $Q_1$ $\mu_{1,1}$ $s_{1,1} $ $n_{1,1}$
$C_1$ $Q_2$ $\mu_{1,2}$ $s_{1,2} $ $n_{1,2}$
... ... ... ... ...
$C_i$ $Q_j$ $\mu_{i,j}$ $s_{i,j} $ $n_{i,j}$

What I want is to calculate the Standard Error of the mean for a specific question. E.e. I want to collect all question ones from all classes and calculate the mean value for $Q_1$.

My guess is that the standard error of the mean for question 1 will be calculated from :

$$SE_{q1} = \sqrt{\frac{s_{1,1}^2}{n_{1,1}} + \frac{s_{2,1}^2}{n_{2,1}} +\ldots +\frac{s_{k,1}^2}{n_{k,1}} } = \sqrt{\sum_{i=1}^k\frac{s_{i,1}^2}{n_{i,1}}} $$

for question J: $$SE_{qJ} = \sqrt{\sum_{i=1}^k\frac{s_{i,J}^2}{n_{i,J}}} $$

So the idea is that the expected mean is the weighted average of the means, plus/minus the above standard error (times the critical T-value).

ie.:

  • the expected mean value $\bar{x}_{J}$:

$$\bar{x}_{J} = \frac{\sum_{i=1}^k \mu_{i,J}\cdot n_{i,J}}{\sum_{i=1}^k n_{i,J}} $$

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    $\begingroup$ "Mean" has several potential interpretations here. Could you describe what property of the population it corresponds to? For instance, it could be the arithmetic mean of responses among the classes; or it could be the arithmetic mean of all student responses; or it could be something else altogether. The first two will differ whenever the classes aren't all the same size. Then, once that is clarified, you need to tell us how you sampled. For instance, is this a stratified sample or is the "class" just one more characteristic of each response? (Non-response is a big issue, so beware!) $\endgroup$
    – whuber ♦
    Commented Jan 8, 2022 at 19:35
  • $\begingroup$ @whuber thank you for your comment. I'll start with the later part, I am ashamed to say that there wasn't any particular sampling scheme. The questionnaires were send to the class population but I had a lot of Non answers, which I "conveniently' decided to consider as a sample. $\endgroup$
    – NMech
    Commented Jan 9, 2022 at 11:58
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    $\begingroup$ @whuber Regarding the first part of your question, If I understand correctly the options you've given, it is the latter (see the definition of the expected mean right at at the end as a weighted mean based on the participants). Essentially, what I am trying to do there is to find a baseline reference of what most people responded in Q1, so that I can later compare between the different classes. $\endgroup$
    – NMech
    Commented Jan 9, 2022 at 12:01

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