must construct a frequency distribution, relative frequency distribution, frequency histograms of a data, relative frequency histogram of the data, describe the

Question

  4.2, 8.6, 9.2, 9.6, 9.6, 9.7, 10.2, 10.5, 10.6, 10.6, 10.9, 10.9, 11.3, 11.4, 11.6, 12.3, 12.6, 13.0, 13.3, 13.4, 13.9, 14.0, 14.3, 14.7, 14.8, 15.5, 15.9, 15.9, 16.1, 16.1, 16.1, 16.2, 17.8, 18.1, 18.3, 18.3, 18.4, 18.4, 18.6, 18.7, 18.9, 19.4, 19.6, 19.7, 20.6, 21.1, 21.2, 21.3, 21.4, 22.2, 25.0 

Answer 1

Quick context: 51 observations. I used Sturges’ rule to create 7 class intervals so the bins are sensible for this sample size.

1) How I chose the bins (step-by-step)

1. Sample size n=51n = 51n=51.
2. Sturges’ rule: k=⌈1+3.322log⁡10(n)⌉k = \lceil 1 + 3.322\log_{10}(n)\rceilk=⌈1+3.322log10​(n)⌉.
3. log⁡10(51)=1.7075701761\log_{10}(51) = 1.7075701761log10​(51)=1.7075701761 (calculated).
4. 3.322×1.7075701761=5.6725481250.3.322\times 1.7075701761 = 5.6725481250.3.322×1.7075701761=5.6725481250.
5. Add 1 → 6.67254812506.67254812506.6725481250.
6. Ceiling → 7 bins.
7. Class width =max−min7=25.0−4.27=20.87≈2.9714.=\dfrac{\text{max} - \text{min}}{7} = \dfrac{25.0 - 4.2}{7} = \dfrac{20.8}{7} \approx 2.9714.=7max−min​=725.0−4.2​=720.8​≈2.9714.

2) Frequency distribution (7 bins — right-inclusive)

Class interval (approx)Frequency
4.179 – 7.171	1
7.171 – 10.143	5
10.143 – 13.114	12
13.114 – 16.086	10
16.086 – 19.057	13
19.057 – 22.029	8
22.029 – 25.000	2

Check: 1+5+12+10+13+8+2=511+5+12+10+13+8+2 = 511+5+12+10+13+8+2=51 (all data included).

3) Relative frequency distribution (proportion and percent)

Class intervalRelative frequency (proportion)Percent
4.179 – 7.171	0.0196	1.96%
7.171 – 10.143	0.0980	9.80%
10.143 – 13.114	0.2353	23.53%
13.114 – 16.086	0.1961	19.61%
16.086 – 19.057	0.2549	25.49%
19.057 – 22.029	0.1569	15.69%
22.029 – 25.000	0.0392	3.92%

Cumulative percent after each bin: 1.96%, 11.76%, 35.29%, 54.90%, 80.39%, 96.08%, 100.00%.

4) ASCII (text) Frequency Histogram — copy/paste friendly

(Each * = 1 observation)

4.179 – 7.171 | *

7.171 –10.143 | *****

10.143–13.114 | ************

13.114–16.086 | **********

16.086–19.057 | *************

19.057–22.029 | ********

22.029–25.000 | **

5) ASCII Relative-Frequency “Histogram” (percent)

(Each # ≈ 2.5% — rough visual)

4.179 – 7.171 | # (1.96%)

7.171 –10.143 | #### (9.80%)

10.143–13.114 | ########## (23.53%)

13.114–16.086 | ######## (19.61%)

16.086–19.057 | ########### (25.49%)

19.057–22.029 | ####### (15.69%)

22.029–25.000 | # (3.92%)

6) Summary statistics (useful for interpretation)

1. n = 51
2. Min = 4.2, Max = 25.0 (range = 20.8)
3. Mean ≈ 15.1765
4. Median = 15.5
5. Sample standard deviation ≈ 4.40
6. Skewness (moment-based) ≈ -0.05 (very close to 0; distribution is approximately symmetric)

7) Plain-language description you can paste directly

The dataset (51 values) ranges from 4.2 to 25.0 and is split into seven equal-width bins. Most values fall in the middle ranges: roughly 59% of the data lie between 10.14 and 19.06 (that’s the three middle bins). The single largest bin is 16.09–19.06 (25.5% of the observations). There’s one small low-value observation near 4.2 and a small handful of higher values near and above 19, but overall the distribution is approximately symmetric and unimodal with its center around 15–16. Typical variability (sample SD) is about 4.4 units.

8) Quick interpretation tips (for students / parents)

1. Center: Around 15–16 (median 15.5).
2. Spread: Most data lie within about ±4.4 of the mean (so many between ~11 and ~19).
3. Outliers: 4.2 is an unusually low observation; 25.0 is a high-value but not extreme relative to the rest.
4. Shape: Single peak (unimodal) and roughly symmetric — no strong long tail to one side.

9) If you want to show work / reproduce this

1. Excel: =FREQUENCY() with bin boundaries (7 bins) or use PivotTable grouping on a numeric column.
2. Google Sheets: same FREQUENCY() + chart → Histogram.
3. Calculator/hand: use Sturges’ rule to pick number of classes (I showed steps), compute class width, then tally counts per class.