Question

I created a data set of final exam scores (out of 100) for 50 students in a statistics class to see how the class performed overall and if the test seemed fair. The scores are: 62, 65, 68, 70, 71, 72, 73, 74, 75, 76, 76, 77, 78, 78, 79, 79, 80, 80, 81, 81, 82, 82, 82, 83, 83, 84, 84, 84, 85, 85, 86, 86, 87, 87, 88, 88, 89, 90, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, and 100. When I look at the distribution, it seems pretty symmetric because most of the scores are around the 80s, and the lower and higher scores spread out in a similar way on both sides, making it look kind of like a bell shape. From that, I estimated the mean to be about 82 since that’s where most of the data is centered, and I estimated the standard deviation to be around 8 to 10 points because most of the scores fall between about 60 and 100. After calculating it more precisely, I got a mean of about 82.7 and a standard deviation of about 9.4. This means the average score was around an 83%, so overall the class did pretty well. The standard deviation shows that most students scored within about 9 points of the average, so the scores aren’t too spread out. Based on this, I think the test was a fair level of difficulty and the results were pretty consistent across the class.


Make the Histogram

Answer 1

I can make a histogram from the scores you gave. First a quick note: the list you pasted contains 49 numbers (not 50), so I used those 49 values. If you actually have a 50th score, tell me and I’ll add it and replot.

Here are the scores I used (sorted, as you gave them):
62, 65, 68, 70, 71, 72, 73, 74, 75, 76, 76, 77, 78, 78, 79, 79, 80, 80, 81, 81, 82, 82, 82, 83, 83, 84, 84, 84, 85, 85, 86, 86, 87, 87, 88, 88, 89, 90, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

ASCII histogram (5-point bins: 60–64, 65–69, …, 95–100). Each '*' = 1 student:

60–64 | *
65–69 | **
70–74 | *****
75–79 | ********
80–84 | ************
85–89 | *********
90–94 | ******
95–100 | ******

Counts by bin: [1, 2, 5, 8, 12, 9, 6, 6] (total 49).

If you’d like a nicer plotted histogram (PNG or interactive), here is ready-to-run Python (matplotlib) code that plots the histogram with the same 5-point bins and also marks the sample mean and ±1 standard deviation lines:

Python (matplotlib) code
- Copy into a .py file or a Jupyter cell and run.

import matplotlib.pyplot as plt
import numpy as np

# replace this list if you have an additional (50th) score
scores = [62,65,68,70,71,72,73,74,75,76,76,77,78,78,79,79,80,80,81,81,82,82,82,83,83,84,84,84,85,85,86,86,87,87,88,88,89,90,90,91,92,93,94,95,96,97,98,99,100]

bins = [60,65,70,75,80,85,90,95,101] # last edge 101 so 100 falls inside final bin
plt.figure(figsize=(8,5))
n, bins_out, patches = plt.hist(scores, bins=bins, edgecolor='black', alpha=0.7)
mean = np.mean(scores)
std = np.std(scores, ddof=1) # sample standard deviation
plt.axvline(mean, color='red', linestyle='-', linewidth=1.5, label=f'Mean = {mean:.2f}')
plt.axvline(mean - std, color='orange', linestyle='--', linewidth=1, label=f'Mean ±1 SD ({mean - std:.2f}, {mean + std:.2f})')
plt.axvline(mean + std, color='orange', linestyle='--', linewidth=1)
plt.xticks(np.arange(60, 101, 5))
plt.xlabel('Score')
plt.ylabel('Number of students')
plt.title('Histogram of Final Exam Scores')
plt.legend()
plt.grid(axis='y', alpha=0.2)
plt.show()

Notes and next steps
- If you want different bin widths (e.g., 10-point bins) or a kernel density overlay, I can modify the code.
- If you want me to produce and attach an image file now, upload the missing score if there is one; otherwise confirm and I’ll generate a PNG and provide a link or base64 image (or give instructions for downloading).