Imagine you are playing your favorite video game and you want to prove that you really are one of the best players in your class. You could say, “I usually score 80 points.” But if that number comes from just one game, would your friends believe you? Probably not! They will want to know: “How many games did you play to get that number?” That question is really about one important idea in statistics: the number of observations.
In statistics, an observation is one piece of data, usually one measurement or one count in a data set. Reporting the number of observations means telling people how many pieces of data you collected. This might sound simple, but it is a powerful part of being clear and honest with data.
As shown in [Figure 1], when we collect data in a table, each row often represents one observation, and counting the rows tells us the number of observations.
An observation is one item in your data set. It can be:
Each time you record a value, you are making an observation. If you collect the math scores for 10 students, you have 10 observations. If your class times how long it takes 15 people to run one lap, you have 15 observations.
We usually write the number of observations with the letter n. For example, if you measured the heights of 12 students, you can write: n = 12.
The number of observations tells other people how big your data set is. This helps them decide how much to trust your conclusions.
For example, if you say “The average score is 85,” that means a lot more if it came from 50 test scores than if it came from only 3 test scores.
Any time we summarize a data set, we should also report how many observations were in that data set. We use it when we talk about:
Whenever you describe data in words or with numbers, the number of observations is part of the story.
Counting observations depends on how your data is shown. Let’s look at a few common forms: lists, tables, and frequency tables.
Suppose you ask 7 friends how many pets they have and you get this list:
\(2, 0, 1, 3, 1, 2, 0\)
Each number is one observation. To find the number of observations, count the numbers in the list.
Now imagine a table of quiz scores:
Student: A, B, C, D, E
Score: 7, 9, 10, 6, 8
Each row (each student) is one observation. Count the rows with data:
So the number of observations is 5. We write: n = 5.
Sometimes we see data in a frequency table, which shows how often each value appears. For example, think about the number of goals scored by a soccer team in 10 games:
Goals: 0, 1, 2, 3
Frequency: 1, 4, 3, 2
This means:
To find the number of observations, we add the frequencies:
\(1 + 4 + 3 + 2 = 10\)
So, n = 10 observations (10 games).
[Figure 2] illustrates how each row in a frequency table contributes to the total number of observations when you add all the frequencies together.
When you share results from data, you should always tell people what you measured and how many observations you had. Good reporting usually includes three pieces:
Here are some clear examples:
Problem: A teacher records the following math quiz scores from one class:
\(6, 8, 10, 7, 9, 8, 5, 10\)
Report the number of observations and describe what it means.
Step 1: Count the observations.
We count how many numbers are in the list:
\(6\) – 1
\(8\) – 2
\(10\) – 3
\(7\) – 4
\(9\) – 5
\(8\) – 6
\(5\) – 7
\(10\) – 8
So there are 8 observations. That means n = 8.
Step 2: Say what the observations represent.
Each observation is one student’s quiz score.
Final report:
“There are n = 8 observations, meaning scores from 8 students.”
Problem: A group collects data on shoe sizes in their class and makes this frequency table:
Shoe Size: 4, 5, 6, 7
Number of Students: 2, 5, 6, 3
Find and report the number of observations.
Step 1: Add the frequencies.
We add the “Number of Students” values:
\(2 + 5 + 6 + 3 = 16\)
So there are 16 observations. That means n = 16.
Step 2: Explain what that means.
There are 16 students whose shoe sizes were recorded.
Final report:
“The data set has n = 16 observations, which are the shoe sizes of 16 students.”
Problem: A student tracks how many hours per day they spend on screens for 5 days:
Day: Mon, Tue, Wed, Thu, Fri
Hours: 3, 2, 4, 3, 5
Report the number of observations and give a good sentence describing the data using that number.
Step 1: Count the observations.
There are 5 days listed, each with a number of hours. So n = 5.
Step 2: Describe the data in context.
You might also want to know the mean to make a better description.
Step 2a: Find the mean (average) screen time.
Add the hours:
\(3 + 2 + 4 + 3 + 5 = 17\)
Divide by the number of observations, which is 5:
\(\textrm{mean} = \frac{17}{5} = 3.4\)
Step 3: Put it all together in a sentence.
“Over 5 days (n = 5), the student spent an average of 3.4 hours per day on screens.”
Often, we use the number of observations together with other statistics. Some common summaries are:
All of these depend on n, the number of observations. For example, to know which value is “in the middle” for the median, we first need to know how many values there are in total.
[Figure 3] shows a dot plot of a small data set with each dot labeled as one observation, and it indicates how counting dots gives the number of observations.
Reporting the number of observations is not just for math class. It is used in many real-life situations that affect people’s decisions.
Sports announcers often say things like, “Her free-throw percentage this season is 82%.” That percentage is based on many free throws, maybe n = 150. If she had only taken 2 shots and made both, you could say 100%, but with n = 2 it would not be very meaningful.
The bigger n is, the more confident people feel that the percentage shows the player’s true skill.
Game companies and app makers collect data on how people use their products. For example, they might say, “The average player spends 40 minutes per day in the game, based on n = 50,000 players.” That large n shows the data is based on a lot of people and is more trustworthy.
When doctors test a new medicine, they run studies with many people. A report might say, “The average decrease in pain was 30%, with n = 2,000 patients.” The number of observations tells us how big the study was. A medicine tested on only 5 people (n = 5) is not as convincing as one tested on thousands.
Principals and teachers look at test scores to see how a grade level is doing. They might say, “The mean reading score for sixth grade was 78, based on n = 120 students.” This shows the data includes all students, not just a few.
Surveys ask people questions, like “Do you think school should start later?” When you hear the results, you should always ask: “How many people answered?” A survey with n = 1,000 responses is usually more reliable than one with n = 20.
Mistake 1: Forgetting to say how many observations there are.
Someone might say, “The average score is 90,” but not tell you that this was from only 2 tests. Always include n.
How to fix it: Get into the habit of writing statements like, “The mean score was 90, with n = 2 tests.”
Mistake 2: Mixing up the number of categories with the number of observations.
For example, in a shoe-size table there might be 4 different sizes, but 16 students. The number of categories is 4. The number of observations is 16.
How to fix it: Remember: categories are types of values; observations are the actual data points. When you add up the frequencies, you get the number of observations.
Mistake 3: Counting missing or blank entries.
If a table has some blank spots (for people who did not answer), you should not count those as observations.
How to fix it: Only count rows that actually have data recorded.
For example:
Using clear, complete sentences like these helps everyone understand not just what your numbers are, but also how much data those numbers are based on. That is the heart of reporting the number of observations.
1. An observation is one data value (like one score, one height, or one time).
2. The number of observations is how many data values are in your set. We usually write it as n.
3. You can find n by:
4. Always report n when you give summaries like mean, median, or range.
5. The bigger n is, the more reliable your summary usually is.
6. Reporting the number of observations helps others understand and trust your data, whether it is about games, sports, school, or real-world studies.
