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AI HL · 13.3

Statistics

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13.3.9

Statistics review: exam questions

13.3.9

Statistics review: exam questions

Target: I can decide whether data is discrete or continuous, and read the total and the mode from a frequency table. (+4 more)

By the end of this section

Key points

Types of data and frequency tables

  • Discrete data is counted: only separate values are possible. Continuous data is measured: any value in a range is possible.
  • The total frequency is the sum of the frequency row. The mode is the value with the highest frequency, not the frequency itself.
  • On the calculator, put the values in one list and the frequencies in another, and use one-variable statistics with the frequency list.
  • Use σ (the population standard deviation) from the calculator.

Box plots and outliers

  • A box plot shows the minimum, Q1, median, Q3 and maximum. IQR = Q3 − Q1. Range = maximum − minimum.
  • Each section of a box plot holds 25% of the values. With 160 days, a quarter is 40 days.
  • Outlier fences: Q1 − 1.5 × IQR and Q3 + 1.5 × IQR. A value outside them is an outlier.
  • On a box plot an outlier is a cross, and the whisker stops at the largest value that is not an outlier.

Sampling, correlation and regression

  • In a stratified sample, the number from each group is (group size ÷ population) × sample size.
  • Pearson's r near 1 or −1 is a strong linear correlation. Say positive or negative, and strong, moderate or weak.
  • The regression line of y on x is used to predict y from x. Predicting outside the range of the data (extrapolation) is unreliable.
  • The gradient is the change in y for each increase of 1 in x. Say it in the context.

On the calculator

Mean and standard deviation from a frequency table

  • Enter the values in one list and the frequencies in another.
  • Choose one-variable statistics and set the frequency list.
  • Read the mean, σ and the total frequency n.

Screenshot and video to come

Quartiles from raw data

  • Enter the data in one list and run one-variable statistics.
  • Read Q1, the median and Q3, then work out the IQR and the two fences by hand.

Screenshot and video to come

Correlation and regression

  • Enter x and y in two lists and run linear regression.
  • Read the gradient, the intercept and r. Write the equation as y = ax + b.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A frequency table

Members of a library were asked how many books they borrowed last week. The results are in the table. (a) State whether the number of books is discrete or continuous. (b) Write down (i) the number of members asked; (ii) the modal number of books. (c) Find (i) the mean number of books; (ii) the standard deviation.

Books borrowed012345Frequency59141273
Working
  1. Type of data

    Discrete

    Books are counted, so only whole numbers are possible.

  2. Total and mode

    (i) 5+9+14+12+7+3 = 50 members. (ii) The highest frequency is 14, so the mode is 2 books.

    The total is the sum of the frequencies. The mode is the value with the highest frequency, not the frequency itself.

  3. Enter the table

    Values in one list, frequencies in another. One-variable statistics with frequency list.

    Tell the calculator which list holds the frequencies.

  4. Read the results

    From the calculator: mean 2.32, standard deviation σ = 1.35

    Use the population standard deviation σ, the one the calculator calls σx. Not the sample one.

Video walkthrough coming
Reading a box plot

The number of visitors to a museum on each of 160 days is shown in the box plot. (a) Write down the median. (b) Write down the minimum. (c) Find the interquartile range. (d) Find the number of days with between 260 and 330 visitors. (e) A storm closed most of the museum on one day and only 25 people visited. Determine whether this day is an outlier.

Visitors120140160180200220240260280300320340360380400420440460480500Visitors
Working
  1. Median and minimum

    Median = 260, minimum = 140

    The median is the line in the box. The minimum is the left end of the whisker.

  2. IQR

    IQR = Q3 − Q1 = 330 − 210 = 120

    Q1 = 210 and Q3 = 330 are the two ends of the box.

  3. Quartiles as days

    Between Q2 and Q3 is 25% of 160 days = 40 days

    Each section of the plot holds a quarter of the days: Q1 to the median, and the median to Q3, each hold 25%.

  4. Outlier fences

    Q1 − 1.5 × IQR = 210 − 1.5 × 120 = 30. Q3 + 1.5 × IQR = 330 + 1.5 × 120 = 510

    A value below the lower fence or above the upper fence is an outlier.

  5. Decide

    25 < 30, so the day is an outlier.

    Always compare with the fence and say so.

Video walkthrough coming
Quartiles and outliers from raw data

The waiting times, in minutes, of 14 patients at a clinic were 12, 8, 15, 22, 9, 31, 14, 11, 17, 10, 13, 16, 9, 58 Find the upper fence for outliers, and state any outliers.

××Wait051015202530354045505560Waiting time (min)
Working
  1. Quartiles

    From the calculator: Q1 = 10, median = 13.5, Q3 = 17

    Enter the data and take one-variable statistics. With 14 values the calculator splits them into two halves of 7.

  2. IQR

    IQR = 17 − 10 = 7

    Upper quartile minus lower quartile.

  3. Fences

    Lower: 10 − 1.5 × 7 = −0.5 Upper: 17 + 1.5 × 7 = 27.5

    The rule is Q1 − 1.5 × IQR and Q3 + 1.5 × IQR.

  4. Outliers

    31 and 58 are above 27.5, so they are outliers. No value is below −0.5.

    The box plot shows an outlier as a cross. The whisker stops at the largest value that is not an outlier.

Video walkthrough coming