Calculate the standard deviation of the given data using the assumed mean method and state the frequency of…

Question

Calculate the standard deviation of the given data using the assumed mean method and state the frequency of the modal class.

ScanSolve AI Tutor · Accepted Answer

To solve this problem, we first need to organize the given data and calculate the midpoints of each class interval. We will also calculate the products fd and fd² for the assumed mean method. Given data: Length (cm) | No. of rods (f) ------------|---------------- 8-10 | 4 11-13 | 7 14-16 | 11 17-19 | 15 20-22 | 8 23-25 | 5 26-28 | 3 Let A = 18 be the assumed mean. We calculate the midpoint (x) for each class, the deviation d = x − A, fd, and fd². Length (cm) — f — Midpoint (x) — d = x − 18 — fd — fd²8-10 — 4 — 9 — −9 — −36 — 324 11-13 — 7 — 12 — −6 — −42 — 252 14-16 — 11 — 15 — −3 — −33 — 99 17-19 — 15 — 18 — 0 — 0 — 0 20-22 — 8 — 21 — 3 — 24 — 72 23-25 — 5 — 24 — 6 — 30 — 180 26-28 — 3 — 27 — 9 — 27 — 243 Total — ∑ f = 53 — ∑ fd = −30 — ∑ fd² = 1170 a) State the frequency of the modal class. Step 1: Identify the modal class. The modal class is the class interval with the highest frequency. From the table, the highest frequency is 15, which corresponds to the class interval 17-19. Step 2: State the frequency. The frequency of the modal class is 15. The frequency of the modal class is 15. b) Using 18 as an assumed mean, calculate: i) Actual mean Step 1: Use the formula for the actual mean with an assumed mean. The formula is x = A + (∑ fd)/(∑ f). Step 2: Substitute the values. x = 18 + (−30)/(53); x = 18 − 0.5660377... x ≈ 17.434 The actual mean is 17.434 cm. ii) Variance Step 1: Use the formula for variance with an assumed mean. The formula is σ² = (∑ fd²)/(∑ f) − ((∑ fd)/(∑ f))². Step 2: Substitute the values. σ² = 1170/53 − ((−30)/(53))²; σ² = 22.075471698 − (−0.5660377358)²; σ² = 22.075471698 − 0.320409009; σ² = 21.755062689 σ² ≈ 21.755 The variance is 21.755 cm². iii) Standard deviation Step 1: Use the formula for standard deviation. The formula is σ = √(σ²). Step 2: Substitute the variance value. σ = √(21.755062689); σ = 4.6642319... σ ≈ 4.664 The standard deviation is 4.664 cm.

Calculate the standard deviation of the given data using the assumed mean method and state the frequency of the modal class.

Answers

Step-by-step solution

Part a· 2 steps

Identify the modal class.

State the frequency.

Part b(i)· 2 steps
$\text{17.434 cm}$

Use the formula for the actual mean with an assumed mean.

Substitute the values.

Part b(ii)· 2 steps
$\text{21.755 cm}^2$

Use the formula for variance with an assumed mean.

Substitute the values.

Part b(iii)· 2 steps
$\text{4.664 cm}$

Use the formula for standard deviation.

Substitute the variance value.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Length (cm)	$f$	Midpoint ( $x$ )	$d = x - 18$	$fd$	$fd^2$
8-10	4	9	-9	-36	324
11-13	7	12	-6	-42	252
14-16	11	15	-3	-33	99
17-19	15	18	0	0	0
20-22	8	21	3	24	72
23-25	5	24	6	30	180
26-28	3	27	9	27	243
Total	$\sum f = 53$			$\sum fd = -30$	$\sum fd^2 = 1170$

Calculate the standard deviation of the given data using the assumed mean method and state the frequency of the modal class.

Answers

Step-by-step solution

Part a· 2 steps

Identify the modal class.

State the frequency.

Part b(i)· 2 steps17.434 cm\text{17.434 cm}17.434 cm

Use the formula for the actual mean with an assumed mean.

Substitute the values.

Part b(ii)· 2 steps21.755 cm2\text{21.755 cm}^221.755 cm2

Use the formula for variance with an assumed mean.

Substitute the values.

Part b(iii)· 2 steps4.664 cm\text{4.664 cm}4.664 cm

Use the formula for standard deviation.

Substitute the variance value.

Try one yourself

Still stuck on this one?

More Mathematics Questions

Part b(i)· 2 steps
$\text{17.434 cm}$

Part b(ii)· 2 steps
$\text{21.755 cm}^2$

Part b(iii)· 2 steps
$\text{4.664 cm}$