Statistics and Data Analysis

Grade 10 · Math · Study Guide

Measures of center and spread, data displays, and interpreting statistical summaries.

Statistics is the science of collecting, analyzing, and interpreting data. At its core, you want to summarize a data set with a few key numbers. The three measures of center — mean, median, and mode — each describe the 'typical' value in different ways. The mean (arithmetic average) adds all values and divides by the count. The median is the middle value when data is sorted. The mode is the most frequent value. The mean is sensitive to outliers; the median is resistant to them.

Measures of spread describe how much the data varies. The range is simply maximum minus minimum. The interquartile range (IQR) is the range of the middle 50%: IQR = Q3 - Q1, where Q1 is the 25th percentile and Q3 is the 75th percentile. The IQR is more robust than the range because it ignores extreme values. Standard deviation measures how far individual values typically stray from the mean — a large standard deviation means high variability.

Box plots (box-and-whisker plots) display the five-number summary: minimum, Q1, median, Q3, maximum. They make it easy to compare distributions visually. Histograms show the shape of a distribution — you look for whether it's symmetric (normal bell curve), skewed left, or skewed right. Skewed right means a long tail to the right; the mean is pulled in that direction.

Scatterplots show the relationship between two numerical variables. The trend can be positive (both go up), negative (one goes up as the other goes down), or no correlation. A line of best fit (regression line) models the trend, and the correlation coefficient r (between -1 and 1) measures the strength and direction.

Key terms

Mean
Sum of all values divided by the count; the arithmetic average.
Median
The middle value in sorted data; resistant to outliers.
IQR (Interquartile Range)
Q3 - Q1; the spread of the middle 50% of data.
Standard Deviation
Average distance of data values from the mean; measures variability.
Correlation Coefficient (r)
A number from -1 to 1 measuring the strength and direction of linear association.

Flashcards

Mean
Sum of values / count of values
Median
Middle value of sorted data; use average of two middle values for even-count sets
Mode
Most frequently occurring value; a dataset can have multiple modes or none
IQR
Interquartile range = Q3 - Q1; spread of the middle 50%
Outlier detection
Values below Q1 - 1.5·IQR or above Q3 + 1.5·IQR are outliers
Correlation coefficient r
r close to ±1: strong linear correlation; r close to 0: weak/no correlation

Practice questions

  1. For the data set {3, 5, 7, 9, 11}, what is the mean?

    1. 5
    2. 6
    3. 7
    4. 9
    Show answer

    Answer: 7

    Mean = (3+5+7+9+11)/5 = 35/5 = 7.

  2. The median is more resistant to outliers than the mean.

    Show answer

    Answer: True

    An outlier shifts the mean significantly but doesn't move the median much. The median only depends on the middle position.

  3. IQR = Q3 - ___.

    Show answer

    Answer: Q1

    IQR = Q3 - Q1. It measures the width of the box in a box plot (the middle 50% of the data).

  4. A distribution that is 'skewed right' has:

    1. A long tail to the left
    2. A long tail to the right
    3. Perfectly equal sides
    4. No mode
    Show answer

    Answer: A long tail to the right

    Skewed right means the tail extends to the right. The mean is pulled toward the tail, making mean > median in right-skewed distributions.

  5. A correlation coefficient of r = -0.9 indicates a strong negative linear relationship.

    Show answer

    Answer: True

    r = -0.9 is close to -1, indicating a strong negative linear relationship — as one variable increases, the other decreases.

Fun facts

Get a personalized course on Statistics and Data Analysis built around your child.