Statistics

Statistics are all around us, from sports averages to news reports about the weather.

• A basketball player averages 14.6 points per game.
• Last year was the hottest year on record since 1897.

Statistics is the science of designing investigations and then collecting, organizing, analyzing, and interpreting data to answer a specific question.

A statistical investigation follows these five key steps:

1. Pose the Question: Define clearly what you want to investigate.
2. Collect Data: Gather information relevant to your question.
3. Analyze Data: Calculate summary statistics, such as frequencies, means, or modes.
4. Represent Data: Display the organized data visually using graphs and charts.
5. Interpret Data: Draw conclusions to answer your question.

A statistical problem is a clear, focused question that can be answered by collecting and analyzing data.

Let's investigate the problem: "What is the favorite school subject of students at our school?"

A population is the entire group of people or things that you want to study.

For our problem, the population is all students at our school.

Data is the information we collect, like numbers, words, measurements, or observations.

The data we collect will be the list of answers from each student, such as "Math", "Art", "Science", "Art", "Sport", etc.

Data can be described by different types of variables:

• Qualitative variable (categorical): Describes a quality or category. The possible answers are words or labels. Think: "What kind?" (e.g., eye color, favorite food).
• Quantitative variable (numerical): Describes a quantity or number that can be counted or measured. Think: "How many?" or "How much?" (e.g., height, number of siblings).

For our survey:

• The variable "Favorite Subject" is qualitative because the answers are categories (Math, Art, etc.).
• The variable "How many hours do you do homework per week?" is quantitative because the answer will be a number that can be measured (e.g., 2 hours, 5 hours).

When we conduct an investigation, we must decide who to collect data from. Do we ask everyone in the population, or just a smaller group?

• A census collects data from every single member of the population. It is accurate but can be very slow and expensive for large populations.
• A survey (or sampling) collects data from a smaller, manageable group taken from the population, called a sample. It is much faster, but the sample must be chosen carefully to be representative of the whole population.

An investigation asks: "What is the favorite subject of students at this school?"
If you ask every student in the school, is it a census or a survey?

An investigation asks: "What is the favorite subject of students at this school?"
If you only ask the students in your math class, is it a census or a survey?

Once you have your question and have identified your population, you need a systematic way to collect and record the answers. For a census of a small group (like your class), a tally table is an efficient tool.

1. Prepare a Tally Table: Before you start, create a table with a column for each possible answer category and a column for tally marks.
2. Conduct the Survey: Ask each person in your population the survey question one by one.
3. Record Each Answer: For each answer you receive, place one tally mark in the row for the corresponding category. Remember to make the fifth mark a diagonal line across the first four to make counting easier.

Collecting data for the question "What is your favorite school subject?"
As you survey each of the 25 students, you would fill out the tally column. After asking all 25 students, your tally table would look like this:

Subject	Tally Marks
Math
Science
Sport
Art

One of the most common ways to collect information about a large group is to use a sample. For a sample to be meaningful, it must fairly represent the entire population. Two key challenges in sampling are avoiding bias and choosing a sample size that is large enough to capture the population’s diversity.

• Selection bias: A famous example of biased sampling is the Literary Digest poll before the 1936 U.S. presidential election. The magazine sent millions of surveys using telephone books and car registration lists. But during the Great Depression, many people couldn’t afford phones or cars. This led to a sample heavily biased toward wealthier citizens, who at the time were more likely to vote Republican. Because of this bias, the poll incorrectly predicted a landslide win for Alfred Landon, while Franklin D. Roosevelt actually won by a wide margin.
• Sample size: During the Cuban Missile Crisis of 1962, U.S. intelligence underestimated the number and types of Soviet missiles in Cuba because they had only a limited amount of reconnaissance data. The small “sample” of photos covered only part of the island, so several launch sites, including some with longer-range missiles, were initially missed. Although this was not a statistical survey, it illustrates how having too little data (a very small or incomplete sample) can lead to serious errors in judgment, especially when the stakes are high.

Sampling error is the natural, expected difference between a statistic calculated from a sample and the true value for the entire population. It exists simply because a sample is a part of the whole, not the whole itself, and because the individuals chosen by chance will never match the population perfectly.

Selection bias is a systematic error that occurs when the method of selecting a sample causes it to be unrepresentative of the population. Unlike random sampling error, bias pushes the results in a specific direction and does not disappear just by increasing the sample size.

An investigation asks: "What is the favorite subject of students at this school?" To collect data, you survey only the students in your math class.
This is a biased sample.

• The Population is all students in the entire school.
• The Sample is only the students in your specific math class.

The results will likely be biased because this sample is not representative of the whole school. For example, students in a math class might be more likely to choose Math as their favorite subject than students selected from an art class or from the general school population. Students from other year levels or special programs may also have very different preferences, but their opinions are completely excluded. Therefore, your conclusion cannot be generalized to the entire school.

To ensure a survey's results are as accurate as possible, statisticians use two main strategies:

1. Use Random Sampling: To avoid selection bias, every member of the population should have an equal chance of being chosen for the sample. This is the foundation of reliable surveying.
2. Increase the Sample Size: As the sample size gets larger, it is more likely to accurately reflect the diversity of the population, and the random sampling error generally decreases. However, increasing the sample size alone does not remove selection bias if the method of selection is already flawed.

Frequency (or count) is how many times each category appears in our data.
Relative frequency is the proportion of the data that falls into a category. We can write it as a fraction, a decimal, or a percentage.$$ \text{Relative frequency (in }\pourcent\text{)} = \frac{\text{Frequency}}{\text{Total number of data values}} \times 100\pourcent $$

Let's calculate the relative frequency for our "Favorite Subject" survey of 25 students:

Subject	Frequency	Relative Frequency
Math	8	\(8/25\times 100\pourcent = 32\pourcent\)
Science	5	\(5/25\times 100\pourcent = 20\pourcent\)
Sport	7	\(7/25\times 100\pourcent = 28\pourcent\)
Art	5	\(5/25\times 100\pourcent = 20\pourcent\)
Total	25	100\(\pourcent\)

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set. The main measures are the mean, median, and mode.

The mode is the value or category that appears most frequently. A dataset can have more than one mode.

The results of the "Favorite Subject" survey are shown in the frequency table below.

Subject	Frequency
Math	8
Science	5
Sport	7
Art	5

What is the mode of this dataset?

The mean is the sum of all numerical values divided by the count of values. It is often called the "average."$$\begin{aligned} \bar{x} &= \frac{\text{sum of all values}}{\text{number of values}} \\ &= \frac{x_1 + x_2 + x_3 + \dots + x_n}{n}\end{aligned}$$

For the dataset \(1, 4, 2, 3, 5, 4, 5, 4, 4\), what is the mean?

The median is the middle value of a dataset that has been ordered from least to greatest.

• If there is an odd number of values, the median is the single middle value.
• If there is an even number of values, there are two middle values and the median is the mean of these two values.

For the dataset \(1, 4, 2, 3, 5, 4, 5, 4, 4\), what is the median?

A measure of spread (or dispersion) describes how varied or "spread out" the values in a dataset are. While central tendency tells us about the center, spread tells us about the consistency of the data. The main measures are the range, interquartile range (IQR), and standard deviation.

Consider the test scores of two students:

• Student A's scores: 10, 50, 90
• Student B's scores: 45, 50, 55

Both students have the same mean score (50), but Student A's scores are very spread out (high dispersion), while Student B's scores are very consistent (low dispersion).

The range is the difference between the maximum and minimum values in a dataset. It gives a quick measure of the total spread.$$ \text{Range} = \text{Maximum Value} - \text{Minimum Value} $$

Find the range for the data: \(1, 19, 10, 2, 18, 10, 5, 15, 10\).

Quartiles are values that divide an ordered dataset into four equal parts.

• The lower quartile (Q1) is the median of the lower half of the data.
• The median (Q2) is the overall median of the data.
• The upper quartile (Q3) is the median of the upper half of the data.

When there is an odd number of values, the median (Q2) is not included in either half when finding Q1 and Q3.
The interquartile range (IQR) is the range of the middle 50\(\pourcent\) of the data. It is less affected by extreme outliers than the range.$$ \text{IQR} = Q_3 - Q_1 $$

Find the quartiles and the interquartile range for the data: \(1, 19, 10, 2, 18, 10, 5, 15, 10\).

The most powerful measure of spread is the standard deviation. It measures the average distance of each data point from the mean. A small standard deviation means the data points are clustered closely around the mean (like Student B), while a large standard deviation means the data is spread out (like Student A).

The standard deviation (\(\sigma\)) is found by taking the square root of the average of the squared differences from the mean.$$\sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n}}$$where \(\bar{x}\) is the mean and \(n\) is the number of data points.

Find the standard deviation for the data: $$2, 4, 4, 4, 5, 5, 7, 9$$

Once data is organized in a table, we can create graphs to see the patterns visually. Bar charts are excellent for comparing frequencies, while pie charts are best for showing proportions.

A bar chart uses rectangular bars where the height of each bar represents its frequency. It can be used for both qualitative and quantitative data.

• For qualitative (categorical) data, each bar represents a distinct category (e.g., "Math", "Science"). The bars are typically drawn with gaps between them to show the categories are separate.
• For quantitative (numerical) data, each bar represents a specific number (e.g., a score of "3", or "4 siblings"). The numbers are placed in order along the horizontal axis.

Draw a bar chart for our "Favorite Subject" survey data.

Subject	Frequency
Math	8
Science	5
Sport	7
Art	5

A pie chart shows the proportion of each category as a slice of a circle. If the relative frequency of a category is written as a decimal (for example \(0.32\)), the angle for its slice is calculated as:$$ \text{Angle} = \text{Relative Frequency} \times 360^\circ. $$

Draw a pie chart for our "Favorite Subject" survey data (Total = 25 students).

Subject	Frequency	Relative Frequency
Math	8	\(32\pourcent\)
Science	5	\(20\pourcent\)
Sport	7	\(28\pourcent\)
Art	5	\(20\pourcent\)

While measures like the mean and standard deviation are powerful, they don't give a full picture of the data's distribution. A box plot (or box-and-whisker plot) is a visual tool that summarizes a dataset by showing both its center and its spread simultaneously.

A box plot visually displays the five-number summary above a number line.

• A central box is drawn from the lower quartile (Q1) to the upper quartile (Q3).
• A line inside the box marks the median (Q2).
• Whiskers (lines) extend from the box out to the minimum and maximum values.

The box represents the middle 50\(\pourcent\) of the data (the IQR), the line inside shows the center, and the whiskers show the full range.

1. Order the data from least to greatest.
2. Find the five-number summary: Calculate the minimum, Q1, median, Q3, and maximum.
3. Draw a number line that covers the full range of your data.
4. Draw the box and median: Draw a box from Q1 to Q3 and a vertical line inside the box at the median.
5. Draw the whiskers: Draw lines from the box to the minimum and maximum values.

This box plot shows the number of minutes passengers spent waiting in an airport departure lounge.Based on the plot, find the median, the range, and the interquartile range.

Interpretation is the final and most important step. It’s where you explain the story your data tells. Follow these steps to write a strong interpretation.

1. State the Main Findings: Start by describing the most obvious points. What is the most popular category (the mode)? What is the least popular?
2. Make Comparisons: Use comparative words (e.g., "more than," "less than," "twice as many") to compare different categories. Perform simple calculations if needed.
3. Use Specific Data as Evidence: Support your statements with numbers. Use frequencies (counts) or relative frequencies (percentages) from your table or chart.
4. Draw a Conclusion: Write a concluding sentence that directly answers your original statistical question.
5. Reflect and Ask New Questions: Think about what your data doesn't tell you. Does this result apply to the whole school, or just your class? What could you investigate next?

Let's apply the 5-step method to interpret the results from our survey on favorite subjects.

Subject	Frequency	Relative Frequency
Math	8	\(32\pourcent\)
Science	5	\(20\pourcent\)
Sport	7	\(28\pourcent\)
Art	5	\(20\pourcent\)

• Main Findings: The data shows that Math is the most popular subject. The least popular subjects are Science and Art, which are tied.
• Comparisons: More students chose Math (8) than Sport (7). The number of students who prefer Science is the same as the number who prefer Art (5).
• Evidence: Math is the favorite for 32\(\pourcent\) of the class, making it the clear mode.
• Conclusion: In conclusion, based on our survey of 25 students, the favorite subject in this class is Math.
• Reflection: This conclusion only applies to our class of 25 students. A new question for a future investigation could be: "Is Math also the favorite subject in the entire school?"

Whenever you encounter a claim based on statistics (e.g., in the news or an advertisement), you must evaluate it critically. A good statistician considers two key points:

1. Question the Data Collection: All sample statistics contain potential error. Always ask:
   • Was the sample representative of the population, or was it biased?
   • Was the sample size large enough to be reliable?
   • What is the potential for sampling error?
2. Avoid Overgeneralization: Statistics describe group trends, not absolute truths about individuals. It is a common misuse of statistics to apply a group average to every member of that group.

• Claim: "Girls are better at math than boys."
• Evidence Provided: In my class, the girls' mean test score was \(87\pourcent\), while the boys' mean was \(75\pourcent\).
• Critical Evaluation: Although the data shows that girls performed better on average in this specific class, the broad generalization is flawed.
• Limitation of the Sample: The sample size (a single class) is too small to be representative of the entire student population. Therefore, this finding cannot be generalized to all girls and boys.
• Corrected Conclusion: A more accurate statement would be: "In this particular class, the girls' average score was higher than the boys' average score."