Statistics II

Slide 1 · Learning Outcomes

What you will be able to do

In Statistics I you found the mean, median and mode. Statistics II adds tools to describe where data lies and how spread out it is: the two numbers every examiner and data analyst asks for.

Calculate the mean of grouped data using an assumed mean.
Draw a cumulative frequency curve (ogive) and read values from it.
Calculate quartiles, deciles and percentiles from grouped data.
Find the range, interquartile range and quartile deviation.
Calculate the variance and standard deviation of ungrouped and grouped data.
Describe the effect on the mean and standard deviation of adding a constant or multiplying by a constant.

Data used in this lesson: marks of 50 Form 4 students in a mathematics test (you can edit the frequencies on slides 2 and 3).

Slide 2 · Mean of grouped data

Using an assumed mean

Choose an assumed mean A (usually a middle mid-point). Let t = x − A. Then:

Mean = A + Σft ÷ Σf

Assumed mean A

Edit any frequency in the table and watch the mean change. Try different values of A: the mean always comes out the same, but the numbers are smaller with a good choice of A.

Slide 3 · Cumulative frequency

The ogive

Add up the frequencies to get the cumulative frequency. Plot each one against the upper class boundary, then join the points with a smooth curve: an ogive.

Percentile50

Green: points joined with straight lines, so the reading matches the formula on the next slide. Dashed grey: the smooth curve you draw in the exam (readings differ very slightly).

Reading the ogive: for the median, go across from n/2 on the cumulative frequency axis to the curve, then down to the marks axis. Lower quartile: n/4. Upper quartile: 3n/4.

Slide 4 · Quartiles, deciles and percentiles

Calculating instead of reading

Value = L + ((position − cf_before) ÷ f) × i

L = lower class boundary of the class containing the position · cf_before = cumulative frequency before that class · f = frequency of that class · i = class width.

Slide 5 · Range and interquartile range

How spread out is the data?

Range

Highest value − lowest value. Easy, but one extreme value (outlier) can make it misleading.

Interquartile range

IQR = Q₃ − Q₁: the spread of the middle 50% of the data. Not affected by extremes.
Quartile deviation (semi-interquartile range) = ½(Q₃ − Q₁).

Activity: compare two classes

Both classes have the same median mark. Which is more consistent?

Slide 6 · Variance and standard deviation

The standard deviation

Variance is the mean of the squared deviations from the mean. Standard deviation (s.d.) is its square root, in the same units as the data.

Variance = Σ(x − x̄)² ÷ n

Variance = Σx² ÷ n − x̄²

Activity: your own data

Type numbers separated by commas

Slide 7 · Standard deviation of grouped data

Grouped data, step by step

Use mid-points x and the assumed mean A with t = x − A:

s.d. = √( Σft² ÷ Σf − (Σft ÷ Σf)² )

Coding shortcut: if all mid-points are equally spaced, use u = (x − A) ÷ i. Then mean = A + i·(Σfu/Σf) and s.d. = i × √(Σfu²/Σf − (Σfu/Σf)²).

Slide 8 · Changing the data

What if every mark changes?

The teacher adds the same bonus to every mark, or scales every mark. What happens to the mean and standard deviation?

Add c0

Multiply by k1

Rules: adding c to every value → mean + c, s.d. unchanged (the spread is the same). Multiplying every value by k → mean × k and s.d. × k (variance × k²).

Slide 9 · Practice

Worked example and quick checks

Worked example

Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

Step 1: Mean = (2+4+4+4+5+5+7+9) ÷ 8 = 40 ÷ 8 = 5

Step 2: Deviations x − 5: −3, −1, −1, −1, 0, 0, 2, 4

Step 3: Squares: 9, 1, 1, 1, 0, 0, 4, 16. Sum = 32. Variance = 32 ÷ 8 = 4

Answer: s.d. = √4 = 2

Your turn

1. Q₁ = 32 and Q₃ = 48. The interquartile range is .

2. The quartile deviation for question 1 is .

3. The variance of a set of data is 25. The standard deviation is .

4. Data has mean 40 and s.d. 6. Every value is increased by 5. The new s.d. is .

Slide 10 · Quiz

Check your understanding

Part A: Fill in the blanks (auto-marked, 1 mark each)

1. A cumulative frequency curve is also called an .

2. Cumulative frequency is plotted against the upper class .

3. The median is the same as the th percentile.

4. The interquartile range is Q₃ minus .

5. The square root of the variance is the standard .

6. Data: 3, 5, 7. The mean is 5, so the variance is (2 d.p.).

7. With assumed mean 45, Σft = −30 and Σf = 50. The mean is .

8. If each value is multiplied by 3, the standard deviation is multiplied by .

9. The ninth decile is the same as the th percentile.

10. Quartile deviation = ½ × .

Part B: Short answers (smart-marked against key points)

Masomotele · Masomotele Technical Training Institute (MTTI), Eldoret · Free KCSE revision