Question

here is a sample data set.
307.1 314.4 316. 321.8 334.9 348.6
352 388.2 408. 422.9 427.2 427.8
429.3 429.3 429. 436.8 445.2 449
453.1 454.9 466.9 466.9 467.4 470.8
488.9 489.1 494.1 494.1 495.6 498.1
504.1 511.7 514.5 520 522.4 523.4
527.4 538.4 550.6 551.5 551.5 551.5
554.9 581 609.9 615.5 620.6 638.7
660.2 661.2 693.9 3 734.8
(histogram: frequency on y - axis, length (cm) on x - axis with bins 300 - 350, 350 - 400, 400 - 450, 450 - 500, 500 - 550, 550 - 600, 600 - 650, 650 - 700, 700 - 750; frequencies as per the bar graph)
find the first quartile for this data set.
$q_1 = \square$
find the third quartile for this data set.
$q_3 = \square$
find the interquartile range for this data set.
$iqr = \square$
find the lower fence separating outliers from usual values.
$\text{fence}_{\text{low}} = \square$
find the upper fence separating outliers from usual values.
$\text{fence}_{\text{high}} = \square$

Explanation:

Step 1: Count the number of data points

First, we count the number of data points. Let's list them:
Row 1: 6 values (307.1, 314.4, 316.?, 321.8, 334.9, 348.6)
Row 2: 6 values (352, 388.2, 408.?, 422.9, 427.2, 427.8)
Row 3: 6 values (429.3, 429.3, 429.3, 436.8, 445.2, 449)
Row 4: 6 values (453.1, 454.9, 466.9, 466.9, 467.4, 470.8)
Row 5: 6 values (488.9, 489.1, 494.1, 494.1, 495.6, 498.1)
Row 6: 6 values (504.1, 511.7, 514.5, 520, 522.4, 523.4)
Row 7: 6 values (527.4, 538.4, 550.6, 551.5, 551.5, 551.5)
Row 8: 6 values (554.9, 581, 609.9, 615.5, 620.6, 638.7)
Row 9: 4 values (660.2, 661.2, 693.9, 734.8) Wait, no, maybe I miscounted. Wait, let's check again. Wait, the first 8 rows have 6 each? Wait, row 9: 660.2, 661.2, 693.9, and then two missing? Wait, maybe the original data has: Let's count properly. Let's list all the numbers:

307.1, 314.4, 316.?, 321.8, 334.9, 348.6,

352, 388.2, 408.?, 422.9, 427.2, 427.8,

429.3, 429.3, 429.3, 436.8, 445.2, 449,

453.1, 454.9, 466.9, 466.9, 467.4, 470.8,

488.9, 489.1, 494.1, 494.1, 495.6, 498.1,

504.1, 511.7, 514.5, 520, 522.4, 523.4,

527.4, 538.4, 550.6, 551.5, 551.5, 551.5,

554.9, 581, 609.9, 615.5, 620.6, 638.7,

660.2, 661.2, 693.9, 734.8. Wait, no, the last row has 4 values? Wait, maybe the original data has 8 rows with 6 each, but the last row has 4? Wait, no, maybe I made a mistake. Let's count the number of data points:

Row 1: 6

Row 2: 6 (total 12)

Row 3: 6 (18)

Row 4: 6 (24)

Row 5: 6 (30)

Row 6: 6 (36)

Row 7: 6 (42)

Row 8: 6 (48)

Row 9: 4? Wait, no, the image shows:

First table:

Row 1: 307.1, 314.4, 316.?, 321.8, 334.9, 348.6 (6)

Row 2: 352, 388.2, 408.?, 422.9, 427.2, 427.8 (6)

Row 3: 429.3, 429.3, 429.3, 436.8, 445.2, 449 (6)

Row 4: 453.1, 454.9, 466.9, 466.9, 467.4, 470.8 (6)

Row 5: 488.9, 489.1, 494.1, 494.1, 495.6, 498.1 (6)

Row 6: 504.1, 511.7, 514.5, 520, 522.4, 523.4 (6)

Row 7: 527.4, 538.4, 550.6, 551.5, 551.5, 551.5 (6)

Row 8: 554.9, 581, 609.9, 615.5, 620.6, 638.7 (6)

Row 9: 660.2, 661.2, 693.9, 734.8 (4) Wait, that's 8*6 +4=52? No, maybe the last row has 6, but some are missing? Wait, the user's image might have a typo, but let's assume the data is sorted. Wait, actually, let's sort the data first. Let's list all the numbers (ignoring the? for now, assuming they are valid):

307.1, 314.4, 316, 321.8, 334.9, 348.6,

352, 388.2, 408, 422.9, 427.2, 427.8,

429.3, 429.3, 429.3, 436.8, 445.2, 449,

453.1, 454.9, 466.9, 466.9, 467.4, 470.8,

488.9, 489.1, 494.1, 494.1, 495.6, 498.1,

504.1, 511.7, 514.5, 520, 522.4, 523.4,

527.4, 538.4, 550.6, 551.5, 551.5, 551.5,

554.9, 581, 609.9, 615.5, 620.6, 638.7,

660.2, 661.2, 693.9, 734.8. Wait, no, that's 8 rows of 6 (48) plus 4? No, maybe the last row has 6, but two are missing (like 708.1 and 715.3? as per the image: 660.2, 661.2, 693.9, 708.1, 715.3, 734.8). Ah, yes, the image shows: 660.2, 661.2, 693.9, 708.1, 715.3, 734.8. So row 9 has 6 values. So total data points: 9 rows *6 =54. Let's confirm:

Row 1:6, Row2:6, Row3:6, Row4:6, Row5:6, Row6:6, Row7:6, Row8:6, Row9:6. Total 54. Good.

So n=54.

Step 2: Find the position of Q1, Q3

For a data set with n observations, the first quartile (Q1) is the value at the position $\frac{n+1}{4}$ (if using the method where we include the median), or $\frac{n}{4}$ (if using the method where we exclude the median). Wait, different methods, but commonly, for n data points, sorted in ascending order, the position of Q1 is at $\frac{n+1}{4}$ th term (Tukey's method) or $\frac{n}{4}$ th term (Moore and McCabe). Let's use the method where we sort the data, then find the median, then split into lower and upper halves.

First, sort the data in ascending order. Let's list all the data points (sorted):

307.1, 314.4, 316, 321.8, 334.9, 348.6,

352, 388.2, 408, 422.9, 427.2, 427.8,

429.3, 429.3, 429.3, 436.8, 445.2, 449,

453.1, 454.9, 466.9, 466.9, 467.4, 470.8,

488.9, 489.1, 494.1, 494.1, 495.6, 498.1,

504.1, 511.7, 514.5, 520, 522.4, 523.4,

527.4, 538.4, 550.6, 551.5, 551.5, 551.5,

554.9, 581, 609.9, 615.5, 620.6, 638.7,

660.2, 661.2, 693.9, 708.1, 715.3, 734.8.

Wait, let's check the sorting. Let's list all values:

307.1,

314.4,

316,

321.8,

334.9,

348.6,

352,

388.2,

408,

422.9,

427.2,

427.8,

429.3,

429.3,

429.3,

436.8,

445.2,

449,

453.1,

454.9,

466.9,

466.9,

467.4,

470.8,

488.9,

489.1,

494.1,

494.1,

495.6,

498.1,

504.1,

511.7,

514.5,

520,

522.4,

523.4,

527.4,

538.4,

550.6,

551.5,

551.5,

551.5,

554.9,

581,

609.9,

615.5,

620.6,

638.7,

660.2,

661.2,

693.9,

708.1,

715.3,

734.8.

Yes, that's 54 values. Now, the median (Q2) is the average of the 27th and 28th values (since n=54, even number). Let's find the 27th and 28th values:

1:307.1, 2:314.4, 3:316, 4:321.8, 5:334.9, 6:348.6,

7:352, 8:388.2, 9:408, 10:422.9, 11:427.2, 12:427.8,

13:429.3, 14:429.3, 15:429.3, 16:436.8, 17:445.2, 18:449,

19:453.1, 20:454.9, 21:466.9, 22:466.9, 23:467.4, 24:470.8,

25:488.9, 26:489.1, 27:494.1, 28:494.1,

So median is (494.1 + 494.1)/2 = 494.1.

Now, the lower half (for Q1) is the first 27 values (since n=54, lower half is 27 values: positions 1 to 27). Wait, no: when n is even, the median is between 27th and 28th, so the lower half is positions 1 to 27, upper half is positions 28 to 54.

So for Q1, we take the median of the lower half (27 values). The median of 27 values is the 14th value (since (27+1)/2 =14). Wait, 27 values: positions 1 to 27. The median is at position (27+1)/2 =14. So the 14th value in the lower half (which is the 14th value overall? Wait no: the lower half is the first 27 values (positions 1-27). So the 14th value in the lower half is the 14th value in positions 1-27. Let's list the lower half (positions 1-27):

1:307.1, 2:314.4, 3:316, 4:321.8, 5:334.9, 6:348.6,

7:352, 8:388.2, 9:408, 10:422.9, 11:427.2, 12:427.8,

13:429.3, 14:429.3, 15:429.3, 16:436.8, 17:445.2, 18:449,

19:453.1, 20:454.9, 21:466.9, 22:466.9, 23:467.4, 24:470.8,

25:488.9, 26:489.1, 27:494.1.

Wait, no, wait: the lower half is positions 1 to 27? Wait, no, when n is even, the median is between 27 and 28, so the lower half is the first 27 values (1-27) and upper half is 28-54 (27 values). So the lower half has 27 values. The median of 27 values is the 14th value (since (27+1)/2=14). So the 14th value in the lower half (positions 1-27) is:

Let's count:

1:307.1

2:314.4

3:316

4:321.8

5:334.9

6:348.6

7:35

Answer:

Step 1: Count the number of data points

First, we count the number of data points. Let's list them:
Row 1: 6 values (307.1, 314.4, 316.?, 321.8, 334.9, 348.6)
Row 2: 6 values (352, 388.2, 408.?, 422.9, 427.2, 427.8)
Row 3: 6 values (429.3, 429.3, 429.3, 436.8, 445.2, 449)
Row 4: 6 values (453.1, 454.9, 466.9, 466.9, 467.4, 470.8)
Row 5: 6 values (488.9, 489.1, 494.1, 494.1, 495.6, 498.1)
Row 6: 6 values (504.1, 511.7, 514.5, 520, 522.4, 523.4)
Row 7: 6 values (527.4, 538.4, 550.6, 551.5, 551.5, 551.5)
Row 8: 6 values (554.9, 581, 609.9, 615.5, 620.6, 638.7)
Row 9: 4 values (660.2, 661.2, 693.9, 734.8) Wait, no, maybe I miscounted. Wait, let's check again. Wait, the first 8 rows have 6 each? Wait, row 9: 660.2, 661.2, 693.9, and then two missing? Wait, maybe the original data has: Let's count properly. Let's list all the numbers:

307.1, 314.4, 316.?, 321.8, 334.9, 348.6,

352, 388.2, 408.?, 422.9, 427.2, 427.8,

429.3, 429.3, 429.3, 436.8, 445.2, 449,

453.1, 454.9, 466.9, 466.9, 467.4, 470.8,

488.9, 489.1, 494.1, 494.1, 495.6, 498.1,

504.1, 511.7, 514.5, 520, 522.4, 523.4,

527.4, 538.4, 550.6, 551.5, 551.5, 551.5,

554.9, 581, 609.9, 615.5, 620.6, 638.7,

660.2, 661.2, 693.9, 734.8. Wait, no, the last row has 4 values? Wait, maybe the original data has 8 rows with 6 each, but the last row has 4? Wait, no, maybe I made a mistake. Let's count the number of data points:

Row 1: 6

Row 2: 6 (total 12)

Row 3: 6 (18)

Row 4: 6 (24)

Row 5: 6 (30)

Row 6: 6 (36)

Row 7: 6 (42)

Row 8: 6 (48)

Row 9: 4? Wait, no, the image shows:

First table:

Row 1: 307.1, 314.4, 316.?, 321.8, 334.9, 348.6 (6)

Row 2: 352, 388.2, 408.?, 422.9, 427.2, 427.8 (6)

Row 3: 429.3, 429.3, 429.3, 436.8, 445.2, 449 (6)

Row 4: 453.1, 454.9, 466.9, 466.9, 467.4, 470.8 (6)

Row 5: 488.9, 489.1, 494.1, 494.1, 495.6, 498.1 (6)

Row 6: 504.1, 511.7, 514.5, 520, 522.4, 523.4 (6)

Row 7: 527.4, 538.4, 550.6, 551.5, 551.5, 551.5 (6)

Row 8: 554.9, 581, 609.9, 615.5, 620.6, 638.7 (6)

Row 9: 660.2, 661.2, 693.9, 734.8 (4) Wait, that's 8*6 +4=52? No, maybe the last row has 6, but some are missing? Wait, the user's image might have a typo, but let's assume the data is sorted. Wait, actually, let's sort the data first. Let's list all the numbers (ignoring the? for now, assuming they are valid):

307.1, 314.4, 316, 321.8, 334.9, 348.6,

352, 388.2, 408, 422.9, 427.2, 427.8,

429.3, 429.3, 429.3, 436.8, 445.2, 449,

453.1, 454.9, 466.9, 466.9, 467.4, 470.8,

488.9, 489.1, 494.1, 494.1, 495.6, 498.1,

504.1, 511.7, 514.5, 520, 522.4, 523.4,

527.4, 538.4, 550.6, 551.5, 551.5, 551.5,

554.9, 581, 609.9, 615.5, 620.6, 638.7,

660.2, 661.2, 693.9, 734.8. Wait, no, that's 8 rows of 6 (48) plus 4? No, maybe the last row has 6, but two are missing (like 708.1 and 715.3? as per the image: 660.2, 661.2, 693.9, 708.1, 715.3, 734.8). Ah, yes, the image shows: 660.2, 661.2, 693.9, 708.1, 715.3, 734.8. So row 9 has 6 values. So total data points: 9 rows *6 =54. Let's confirm:

Row 1:6, Row2:6, Row3:6, Row4:6, Row5:6, Row6:6, Row7:6, Row8:6, Row9:6. Total 54. Good.

So n=54.

Step 2: Find the position of Q1, Q3

For a data set with n observations, the first quartile (Q1) is the value at the position $\frac{n+1}{4}$ (if using the method where we include the median), or $\frac{n}{4}$ (if using the method where we exclude the median). Wait, different methods, but commonly, for n data points, sorted in ascending order, the position of Q1 is at $\frac{n+1}{4}$ th term (Tukey's method) or $\frac{n}{4}$ th term (Moore and McCabe). Let's use the method where we sort the data, then find the median, then split into lower and upper halves.

First, sort the data in ascending order. Let's list all the data points (sorted):

307.1, 314.4, 316, 321.8, 334.9, 348.6,

352, 388.2, 408, 422.9, 427.2, 427.8,

429.3, 429.3, 429.3, 436.8, 445.2, 449,

453.1, 454.9, 466.9, 466.9, 467.4, 470.8,

488.9, 489.1, 494.1, 494.1, 495.6, 498.1,

504.1, 511.7, 514.5, 520, 522.4, 523.4,

527.4, 538.4, 550.6, 551.5, 551.5, 551.5,

554.9, 581, 609.9, 615.5, 620.6, 638.7,

660.2, 661.2, 693.9, 708.1, 715.3, 734.8.

Wait, let's check the sorting. Let's list all values:

307.1,

314.4,

316,

321.8,

334.9,

348.6,

352,

388.2,

408,

422.9,

427.2,

427.8,

429.3,

429.3,

429.3,

436.8,

445.2,

449,

453.1,

454.9,

466.9,

466.9,

467.4,

470.8,

488.9,

489.1,

494.1,

494.1,

495.6,

498.1,

504.1,

511.7,

514.5,

520,

522.4,

523.4,

527.4,

538.4,

550.6,

551.5,

551.5,

551.5,

554.9,

581,

609.9,

615.5,

620.6,

638.7,

660.2,

661.2,

693.9,

708.1,

715.3,

734.8.

Yes, that's 54 values. Now, the median (Q2) is the average of the 27th and 28th values (since n=54, even number). Let's find the 27th and 28th values:

1:307.1, 2:314.4, 3:316, 4:321.8, 5:334.9, 6:348.6,

7:352, 8:388.2, 9:408, 10:422.9, 11:427.2, 12:427.8,

13:429.3, 14:429.3, 15:429.3, 16:436.8, 17:445.2, 18:449,

19:453.1, 20:454.9, 21:466.9, 22:466.9, 23:467.4, 24:470.8,

25:488.9, 26:489.1, 27:494.1, 28:494.1,

So median is (494.1 + 494.1)/2 = 494.1.

Now, the lower half (for Q1) is the first 27 values (since n=54, lower half is 27 values: positions 1 to 27). Wait, no: when n is even, the median is between 27th and 28th, so the lower half is positions 1 to 27, upper half is positions 28 to 54.

So for Q1, we take the median of the lower half (27 values). The median of 27 values is the 14th value (since (27+1)/2 =14). Wait, 27 values: positions 1 to 27. The median is at position (27+1)/2 =14. So the 14th value in the lower half (which is the 14th value overall? Wait no: the lower half is the first 27 values (positions 1-27). So the 14th value in the lower half is the 14th value in positions 1-27. Let's list the lower half (positions 1-27):

1:307.1, 2:314.4, 3:316, 4:321.8, 5:334.9, 6:348.6,

7:352, 8:388.2, 9:408, 10:422.9, 11:427.2, 12:427.8,

13:429.3, 14:429.3, 15:429.3, 16:436.8, 17:445.2, 18:449,

19:453.1, 20:454.9, 21:466.9, 22:466.9, 23:467.4, 24:470.8,

25:488.9, 26:489.1, 27:494.1.

Wait, no, wait: the lower half is positions 1 to 27? Wait, no, when n is even, the median is between 27 and 28, so the lower half is the first 27 values (1-27) and upper half is 28-54 (27 values). So the lower half has 27 values. The median of 27 values is the 14th value (since (27+1)/2=14). So the 14th value in the lower half (positions 1-27) is:

Let's count:

1:307.1

2:314.4

3:316

4:321.8

5:334.9

6:348.6

7:35