Chapter focus
Data points become useful when they are organized, represented and interpreted. A good statistical statement answers a question using evidence and acknowledges variation in the data.
These original notes support the named Ganita Prakash Part II chapter. Use the assigned textbook for its precise data investigations and activities.
1. Ask a statistical question
A statistical question expects answers that vary across people or observations, such as “How long does each student take to reach school?” A question such as “How many days are in a week?” has one fixed answer and is not statistical.
Collect data consistently. Record units, time period and who or what was measured. A data set without context can be misleading.
2. Organize and represent data
Use a frequency table to count how often each value appears. A bar graph is suitable for comparing categories; label both axes, use equal intervals and give the graph a title. A line graph is useful when values change over time.
Check that the scale starts where intended and that categories are not omitted. A distorted scale can exaggerate small differences.
3. Find representative values
- Mean: sum of all values divided by the number of values.
- Median: middle value after arranging the data in order; for an even count, average the two middle values.
- Mode: most frequent value; a data set may have more than one mode or no mode.
Worked example: 2, 3, 3, 4, 8
Mean = (2 + 3 + 3 + 4 + 8) ÷ 5 = 20 ÷ 5 = 4. Median = 3, the middle value. Mode = 3, the most frequent value.
The large value 8 pulls the mean upward; the median still describes the central position. Choose a representative value that suits the question.
4. Interpret rather than overclaim
Describe what the data show and how many observations support the statement. A class survey of 20 students does not automatically represent every student in a town. Association between two patterns does not by itself prove that one caused the other.
Look for unusual values and missing responses. Explain whether they change the mean, median or conclusion.
Practice: analyze the data
1. Find the mean, median and mode of 2, 3, 3, 4, 5, 3.
Answer: Sum = 20, so mean = 20/6 = 3 1/3. Ordered data are 2, 3, 3, 3, 4, 5; median = 3; mode = 3.
2. Find the median of 4, 6, 7, 10.
Answer: The middle values are 6 and 7, so median = (6 + 7)/2 = 6.5.
3. Which measure is least affected by one unusually large value: mean or median?
Answer: The median is usually less affected because it depends on position after sorting, not the size of every value.
4. A bar graph compares weekly reading times. What must its axes and scale show?
Answer: One axis should identify the categories or weeks, the other reading-time values with units; use a clear title and equal intervals.
5. Why should a survey state whom it asked and when?
Answer: The group and time period define what the data represent and limit what conclusions can reasonably be drawn.
Chapter summary
Statistical reasoning starts with a clear question and reliable data. Organize and graph observations accurately, use mean, median or mode thoughtfully, and keep conclusions within the limits of the sample.