LemonmathSelf-study sheet

Grade 12 · Lesson 3.2

Variance and standard deviation of grouped data

Name: Date: Time spent: min

Goals

Find the variance $s^2$ and the standard deviation $s$ of grouped data with either formula.
Set out the work in a table with the columns $x_i$ , $n_i$ , $n_ix_i$ , $n_ix_i^2$ .
Compare the spread of two sets of grouped data using the standard deviation.
Understand the effect of the class width and of changing units.

Remember

Replace each class by its midpoint $x_i$ with frequency $n_i$ .
$s^2=\dfrac1n\sum n_i(x_i-\bar{x})^2=\dfrac1n\sum n_ix_i^2-\bar{x}^2$ ; $s=\sqrt{s^2}$ , in the same unit as the data.
Draw up a table of $x_i$ , $n_i$ , $n_ix_i$ , $n_ix_i^2$ ; round only the final result.
For two data sets with similar means: smaller $s$ means more even.
The $s$ of grouped data is an approximate value; narrower classes usually give a better approximation.

Exercises

Exercise 1

The daily phone use (hours) of $20$ students is given in the table. Find $s^2$ and $s$ with the definition.

Time (hours)	$[0, 4)$	$[4, 8)$	$[8, 12)$	$[12, 16)$
Students	$3$	$7$	$7$	$3$

Exercise 2

The test marks of $40$ students are grouped as follows. Draw up the table and find $\bar{x}$ , $s^2$ , $s$ .

Mark	$[4, 5)$	$[5, 6)$	$[6, 7)$	$[7, 8)$	$[8, 9)$	$[9, 10)$
Students	$2$	$5$	$9$	$14$	$7$	$3$

Exercise 3

Two delivery services P and Q recorded the delivery times (minutes) of $40$ orders each.

Time (minutes)	$[20, 25)$	$[25, 30)$	$[30, 35)$	$[35, 40)$	$[40, 45)$
Service P	$2$	$10$	$16$	$10$	$2$
Service Q	$8$	$8$	$8$	$8$	$8$

Find the mean time and the standard deviation for each service. Which service delivers more reliably on time?

Exercise 4

Consider the following grouped data.

Class	$[0, 10)$	$[10, 20)$	$[20, 30)$	$[30, 40)$
Frequency	$1$	$4$	$4$	$1$

Is each claim true or false?

a) The mean is $20$ .

b) The variance is $65$ .

c) The standard deviation is greater than $9$ .

d) If every value is increased by $5$ (the classes become $[5, 15)$ , …, $[35, 45)$ ), the variance becomes $70$ .

Exercise 5

The heights (cm) of $50$ students are given in the table. Find the standard deviation (in cm, to two decimal places).

Height (cm)	$[150, 155)$	$[155, 160)$	$[160, 165)$	$[165, 170)$	$[170, 175)$
Students	$5$	$12$	$18$	$10$	$5$

Exercise 6

(Revisiting the mean of grouped data, grade 11.) The times (minutes) that $20$ students took to finish an assignment are given in the table. Find $\bar{x}$ , then $s$ .

Time (minutes)	$[10, 20)$	$[20, 30)$	$[30, 40)$	$[40, 50)$
Students	$3$	$9$	$6$	$2$

Exercise 7

(Revisiting Range and interquartile range of grouped data.) For the two delivery services in Exercise 3, find the interquartile range of each. Is the conclusion the same as with the standard deviation?

Exercise 8 Challenge

Grouped data have the following frequency table, where $x$ is a positive integer. The mean is $4.2$ . Find $x$ , then the variance (to two decimal places).

Class	$[0, 2)$	$[2, 4)$	$[4, 6)$	$[6, 8)$
Frequency	$2$	$x$	$5$	$3$

Exercise 9 Challenge

In Example 1 the sleep data have $s^2=1.1275$ (unit: hours²). If all the data are converted to minutes (every value and every class end multiplied by $60$ ), what are the new variance and standard deviation?

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. In grouped data, the class midpoint of the class

[20, 30)

20

.B.

25

.C.

30

.D.

10

Question 2. A set of grouped data has variance

s^2=2.25

. Its standard deviation is

1.125

.B.

5.0625

.C.

2.25

.D.

1.5

Question 3.

The variance of the following grouped data is

Class	$[0, 2)$	$[2, 4)$	$[4, 6)$	$[6, 8)$
Frequency	$2$	$3$	$3$	$2$

4.2

.B.

5

.C.

2.05

.D.

0

Question 4. Two sets of grouped data

A

and

B

have the same mean, with standard deviations

s_A=2.1

and

s_B=3.4

. Which statement is correct?

A

certainly has a smaller range than

B

.B. The data in

B

are more even than the data in

A

.C. The data in

A

are more even than the data in

B

.D. The two data sets have the same spread.

Part II. True or false

Question 1.

The weekly running distances (km) of $20$ people are given in the table.

Distance (km)	$[10, 20)$	$[20, 30)$	$[30, 40)$	$[40, 50)$
People	$4$	$6$	$8$	$2$

a) The class midpoint of the class $[30, 40)$ is $35$ .	T / F
b) The mean of the data is $29$ km.	T / F
c) The variance of the data is $84$ .	T / F
d) If every value is converted to a different unit by multiplying by $2$ , the new variance is $168$ .	T / F

Part III. Short answer

Question 1.

The weekly private study times (hours) of $30$ students are given in the table. Find the standard deviation of the data, giving your answer to 2 decimal places.

Time (hours)	$[0, 3)$	$[3, 6)$	$[6, 9)$	$[9, 12)$
Students	$4$	$10$	$12$	$4$

Question 2.

The lengths (mm) of $40$ machine parts are given in the table. Find the variance of the data, giving your answer to 2 decimal places.

Length (mm)	$[48, 49)$	$[49, 50)$	$[50, 51)$	$[51, 52)$
Parts	$4$	$16$	$16$	$4$

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $s^2=\dfrac1n\sum n_i(x_i-\bar{x})^2$ with $x_i$ the midpoints.

Steps:

Midpoints $2$ , $6$ , $10$ , $14$ ; find $\bar{x}$ .
Find the deviations, square them and multiply by the frequencies.

Exercise 2.

What you need: $s^2=\dfrac1n\sum n_ix_i^2-\bar{x}^2$ .

Steps:

Midpoints $4.5$ , $5.5$ , …, $9.5$ .
Find $\sum n_ix_i$ and $\sum n_ix_i^2$ .

Exercise 3.

What you need: with equal means, the smaller $s$ means more reliable.

Steps:

Midpoints $22.5$ , $27.5$ , …, $42.5$ . Both tables are symmetric about $32.5$ .
The deviations are $-10$ , $-5$ , $0$ , $5$ , $10$ .

Exercise 4.

What you need: the variance formula; adding the same number to every value leaves the deviations unchanged.

Steps:

Midpoints $5$ , $15$ , $25$ , $35$ ; find $\bar{x}$ .
The deviations are $-15$ , $-5$ , $5$ , $15$ .

Exercise 5.

What you need: $s^2=\dfrac1n\sum n_ix_i^2-\bar{x}^2$ ; do not round $\bar{x}$ along the way.

Steps:

Midpoints $152.5$ , $157.5$ , …, $172.5$ .
Find $\bar{x}$ , then $s^2$ ; take the square root and round only at the end.

Exercise 6.

What you need: $\bar{x}=\dfrac{\sum n_ix_i}{n}$ with $x_i$ the midpoints; $s^2=\dfrac{\sum n_ix_i^2}{n}-\bar{x}^2$ .

Steps:

Midpoints $15$ , $25$ , $35$ , $45$ .
Find $\sum n_ix_i$ and $\sum n_ix_i^2$ .

Exercise 7.

What you need: $Q_1=u_m+\dfrac{\frac n4-C}{n_m}(u_{m+1}-u_m)$ , and likewise for $Q_3$ ; positions $10$ and $30$ .

Steps:

Cumulative frequencies for P: $2$ , $12$ , $28$ , $38$ , $40$ ; for Q: $8$ , $16$ , $24$ , $32$ , $40$ .
Find $Q_1$ , $Q_3$ for each service.

Exercise 8.

What you need: $\bar{x}=\dfrac{\sum n_ix_i}{n}$ with $n=10+x$ .

Steps:

Midpoints $1$ , $3$ , $5$ , $7$ ; set up the equation $\dfrac{48+3x}{10+x}=4.2$ .
With that $x$ , find $\sum n_ix_i^2$ and $s^2$ .

Exercise 9.

What you need: multiplying every value by $k$ multiplies $\bar{x}$ and every deviation by $k$ .

Steps:

Each deviation $x_i-\bar{x}$ is multiplied by $60$ , so its square is multiplied by $3600$ .
Deduce the new $s^2$ , then the new $s$ .

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