Although there is a common tip in critical thinking classes that manipulating the Y-axis range can lead to misleading presentation of a difference, I believe in this particular graph, which clearly provides numbers to compare, you can’t say it is misleading.
People can read and compare the values and draw their own conclusions. And I am saying that without any consideration of the distros discussed, since I am impartial to distros, I like all distros I have tried.
This “study” almost certainly must have way deeper assumptions- and metrics- related problems to start with, so even finding myself having this argument is preposterous. But I am just pointing out the misapplication of critical thinking guideline, and this is a valid point which I insist everyone who relies on to consider, if you care about critical thinking at all.
No one said you are doing layman statistics, the pasted comment is from another discussion, provided here for context, and for very good reasons. It aligns with obvious misconceptions about statistics that should be pointed out. Probability and statistics are thorny subjects that nonetheless are inevitable in order to understand the world surrounding us, material, social, and economic, so yes I will nitpick here and call out the misapplication of canned critical thinking thought-terminating cliches.
There’s not a lot of data to work with, and the kind of test used to determine significance is not the same across the board, but in this case you can do an analysis of variance. Start with a null hypothesis that the happiness level between distros are insignificant, and the alternative hypothesis is that they’re not. Here are the assumptions we have to make:
An alpha value of 0.05. This is somewhat arbitrary, but 5% is the go-to threshold for statistical significance.
A reasonable sample size of users tested for happiness, we’ll go with 100 for each distro.
A standard deviation between users in distro groups. This is really hard to know without seeing more data, but as long as the sample size was large enough and in a normal distribution, we can reasonably assume s = 0.5 for this.
We can start with the total mean, this is pretty simple:
This makes a total sum of squares of 0.471. With our sample size of 100, this makes for a sum of squares between groups of 47.1. The degrees of freedom for between groups is one less than the number of groups (df1 = 10).
The sum of squares within groups is where it gets tricky, but using our assumptions, it would be:
number of groups * (sample size - 1) * (standard deviation)^2
Which calculates as:
11 * (100 - 1) * (0.5)^2 = 272.25
The degrees of freedom for this would be the number of groups subtracted from the sum of sample sizes for every group (df2 = 1089)
Now we can calculate the mean squares, which is generally the quotient of the sum of squares and the degrees of freedom:
# MS (between)47.1 / 10 = 4.71// Doesn't end up making a difference, but just for clarity# MS (within)272.25 / 1089 = 0.25
Now the F-statistic value is determined as the quotient between these:
F = 4.71 / 0.25 = 18.84
To not bog this down even further, we can use an F-distribution table with the following calculated values:
df1 = 10
df2 = 1089
F = 18.84
alpha = 0.05
According to the linked table, the F-critical value is between 1.9105 and 1.8307. The calculated F-statistic value is higher than the critical value, which is our indication to reject the null hypothesis and conclude that there is a statistical significance between these values.
However, again you can see above just how many assumptions we had to make, that the distribution of the data within each group was great in number and normally varied. There’s just not enough data to really be sure of any of what I just did above, so the only thing we have to rely on is the representation of the data we do have. Regardless of the intentions of whoever created this graph, the graph itself is in fact misrepresent the data by excluding the commonality between groups to affect our perception of scale. There’s a clip I made of a great example of this:
There’s a pile of reasons this graph is terrible, awful, no good. However, it’s that scale of the y-axis I want to focus on.
This is an egregious example of this kind of statistical manipulation for the point of demonstration. In another comment I ended up recreating this bar graph with a more proper scale, which has a lower bound of 0 as it should. It’s suggested that these are values out of 10, so that should be the upper bound as well. That results in something that looks like this:
In fact, if you wanted you could go the other way and manipulate data in favor of making something look more insignificant by choosing a ridiculously high upper bound, like this:
But using the proper scale, it’s still quite difficult to tell. If these numbers were something like average reviews of products, it would be easy in that perspective to imagine these as insignificant, like people are mostly just rating 7/10 across the board. However, it’s the fact that these are Linux users that makes you imagine that the threshold for the differences are much lower, because there just aren’t that many Linux users, and opinions wildly vary between them. This also calls into question how that data was collected, which would require knowing how the question was asked, and how users were polled or tested to eliminate the possibility of confounding variables. At the end of the day I just really could not tell visually if it’s significant or not, but that graph is not a helpful way to represent it. In fact, I think Excel might be to blame for this kind of mistake happening more commonly, when I created the graph it defaulted the lower bound to 6. I hope this was helpful, it took me way too much time to write 😂
Oh sport, and I thought I was the one beating on a dead horse here. I understand why people claim to take issue with the Y-axis range. I am just saying chart makers can zoom in to make a point, and it is not automatically misleading. That is all. Anyway, thanks for writing this. Looks like a lot of effort, and some of it will make sense in my stats coursework, thanks!
I am not trying to apply a “critical thinking guideline” I saw elsewhere. I’ve not taken any “critical thinking classes”. I’m more insulted that you think I couldn’t have possibly just thought of that comment myself. It’s not a particularly crazy comment to make, and I don’t see why any individual who knows how to read graphs couldn’t just happen to make that comment.
Anyway—sure, I never said the graph lied. Perhaps a better wording would be that, regardless of how the information is presented, I don’t think the difference in magnitude between people’s happiness ratings (ignoring the issues with how those ratings were collected and ascertained in the first place) is significant or particularly of note. The Y-axis is chosen so as to visually amplify this difference. I didn’t claim the data presented by the graph was untrue or that reading the graph correctly was too difficult if one wanted to read it properly.
I really did not mean to be insulting. I am just saying chart makers can choose to make a zoom in, and it is not automatically propaganda or something. All this has led people astray of the real issues, like WTF is measuring ‘happiness’ on a 1-10 scale, and what are the metric properties of this 1-10 scale. Then there are all the sampling issues and what have you. I just expected more people discussing this stuff rather than the Y-axis.
I didn’t say it was propaganda—the content of the graph reads as quite clearly silly to me and not trying to make a particularly serious or scientific point. I guess the same reason is why I pointed out the Y-axis instead of the sampling issues, because the sampling issues seem much more self-evident.
Although there is a common tip in critical thinking classes that manipulating the Y-axis range can lead to misleading presentation of a difference, I believe in this particular graph, which clearly provides numbers to compare, you can’t say it is misleading.
People can read and compare the values and draw their own conclusions. And I am saying that without any consideration of the distros discussed, since I am impartial to distros, I like all distros I have tried.
This “study” almost certainly must have way deeper assumptions- and metrics- related problems to start with, so even finding myself having this argument is preposterous. But I am just pointing out the misapplication of critical thinking guideline, and this is a valid point which I insist everyone who relies on to consider, if you care about critical thinking at all.
No one said you are doing layman statistics, the pasted comment is from another discussion, provided here for context, and for very good reasons. It aligns with obvious misconceptions about statistics that should be pointed out. Probability and statistics are thorny subjects that nonetheless are inevitable in order to understand the world surrounding us, material, social, and economic, so yes I will nitpick here and call out the misapplication of canned critical thinking thought-terminating cliches.
There’s not a lot of data to work with, and the kind of test used to determine significance is not the same across the board, but in this case you can do an analysis of variance. Start with a null hypothesis that the happiness level between distros are insignificant, and the alternative hypothesis is that they’re not. Here are the assumptions we have to make:
We can start with the total mean, this is pretty simple:
Now we need the total sum of squares, the squared differences between each individual value and the overall mean:
Arch: (6.51 - 6.897)^2 = 0.150 Fedora: (6.71 - 6.897)^2 = 0.035 Mint: (6.74 - 6.897)^2 = 0.025 openSUSE: (6.76 - 6.897)^2 = 0.019 Manjaro: (6.83 - 6.897)^2 = 0.005 Ubuntu: (6.9 - 6.897)^2 = 0.00001 Debian: (6.93 - 6.897)^2 = 0.001 MX Linux: (7 - 6.897)^2 = 0.011 Gentoo: (7.11 - 6.897)^2 = 0.045 Pop!_OS: (7.12 - 6.897)^2 = 0.050 Slackware: (7.26 - 6.897)^2 = 0.132This makes a total sum of squares of 0.471. With our sample size of 100, this makes for a sum of squares between groups of 47.1. The degrees of freedom for between groups is one less than the number of groups (
df1 = 10).The sum of squares within groups is where it gets tricky, but using our assumptions, it would be:
number of groups * (sample size - 1) * (standard deviation)^2Which calculates as:
The degrees of freedom for this would be the number of groups subtracted from the sum of sample sizes for every group (
df2 = 1089)Now we can calculate the mean squares, which is generally the quotient of the sum of squares and the degrees of freedom:
# MS (between) 47.1 / 10 = 4.71 // Doesn't end up making a difference, but just for clarity # MS (within) 272.25 / 1089 = 0.25Now the F-statistic value is determined as the quotient between these:
F = 4.71 / 0.25 = 18.84To not bog this down even further, we can use an F-distribution table with the following calculated values:
According to the linked table, the F-critical value is between 1.9105 and 1.8307. The calculated F-statistic value is higher than the critical value, which is our indication to reject the null hypothesis and conclude that there is a statistical significance between these values.
However, again you can see above just how many assumptions we had to make, that the distribution of the data within each group was great in number and normally varied. There’s just not enough data to really be sure of any of what I just did above, so the only thing we have to rely on is the representation of the data we do have. Regardless of the intentions of whoever created this graph, the graph itself is in fact misrepresent the data by excluding the commonality between groups to affect our perception of scale. There’s a clip I made of a great example of this:
There’s a pile of reasons this graph is terrible, awful, no good. However, it’s that scale of the y-axis I want to focus on.
This is an egregious example of this kind of statistical manipulation for the point of demonstration. In another comment I ended up recreating this bar graph with a more proper scale, which has a lower bound of 0 as it should. It’s suggested that these are values out of 10, so that should be the upper bound as well. That results in something that looks like this:
In fact, if you wanted you could go the other way and manipulate data in favor of making something look more insignificant by choosing a ridiculously high upper bound, like this:
But using the proper scale, it’s still quite difficult to tell. If these numbers were something like average reviews of products, it would be easy in that perspective to imagine these as insignificant, like people are mostly just rating 7/10 across the board. However, it’s the fact that these are Linux users that makes you imagine that the threshold for the differences are much lower, because there just aren’t that many Linux users, and opinions wildly vary between them. This also calls into question how that data was collected, which would require knowing how the question was asked, and how users were polled or tested to eliminate the possibility of confounding variables. At the end of the day I just really could not tell visually if it’s significant or not, but that graph is not a helpful way to represent it. In fact, I think Excel might be to blame for this kind of mistake happening more commonly, when I created the graph it defaulted the lower bound to 6. I hope this was helpful, it took me way too much time to write 😂
Oh sport, and I thought I was the one beating on a dead horse here. I understand why people claim to take issue with the Y-axis range. I am just saying chart makers can zoom in to make a point, and it is not automatically misleading. That is all. Anyway, thanks for writing this. Looks like a lot of effort, and some of it will make sense in my stats coursework, thanks!
I am not trying to apply a “critical thinking guideline” I saw elsewhere. I’ve not taken any “critical thinking classes”. I’m more insulted that you think I couldn’t have possibly just thought of that comment myself. It’s not a particularly crazy comment to make, and I don’t see why any individual who knows how to read graphs couldn’t just happen to make that comment.
Anyway—sure, I never said the graph lied. Perhaps a better wording would be that, regardless of how the information is presented, I don’t think the difference in magnitude between people’s happiness ratings (ignoring the issues with how those ratings were collected and ascertained in the first place) is significant or particularly of note. The Y-axis is chosen so as to visually amplify this difference. I didn’t claim the data presented by the graph was untrue or that reading the graph correctly was too difficult if one wanted to read it properly.
I really did not mean to be insulting. I am just saying chart makers can choose to make a zoom in, and it is not automatically propaganda or something. All this has led people astray of the real issues, like WTF is measuring ‘happiness’ on a 1-10 scale, and what are the metric properties of this 1-10 scale. Then there are all the sampling issues and what have you. I just expected more people discussing this stuff rather than the Y-axis.
I didn’t say it was propaganda—the content of the graph reads as quite clearly silly to me and not trying to make a particularly serious or scientific point. I guess the same reason is why I pointed out the Y-axis instead of the sampling issues, because the sampling issues seem much more self-evident.