版權

1

2

版權

©2019 本書版權屬香港特別行政區政府教育局所有。本書任何 部分之文字及圖片等，如未獲版權持有人之書面同意，不得用 任何方式抄襲、節錄或翻印作商業用途，亦不得以任何方式透 過互聯網發放。

ISBN 978-988-8370-88-7

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編者的話

為配合香港數學教育的發展，並向教師提供更多的參考資料，

課程發展處數學教育組於 2007 年開始邀請大學學者及資深教 師撰寫專文，以及蒐集及整理講座資料，輯錄成《數學百子櫃 系列》 。本書《2018/19 中學生統計創意寫作比賽作品集》 ，是這 個系列的第二十六冊。本書輯錄的文章，大部分是「2018/19 中 學生統計創意寫作比賽」的優勝作品，由參賽的中學生撰寫。

本書所輯錄的參賽作品嘗試透過統計創意寫作，以簡潔的語言 輕鬆地介紹統計的知識。

本書共有 14 篇文章，第 1 至 10 篇為「2018/19 中學生統計 創意寫作比賽」的冠軍、亞軍、季軍和優異作品。其餘 4 篇則 為邀請作品，分別由政府統計處的統計師，香港大學統計及精 算學系的教授和中學教師撰寫，供讀者們閱覽。本書的文章充 滿趣味，期望讀者閱後能獲得啟發、不僅增加統計的知識，還 能善用統計決策、解難。

此書得以順利出版，實有賴這次比賽的籌備委員會成員所付出

的努力。在此，謹向撰寫作品的得獎隊伍、政府統計處的統計

師、香港大學精算及統計學系的教授和朱吉樑老師致以衷心的

4

感謝。最後，更要多謝這次比賽的籌備委員會主席楊良河博士 和總評審主任張家俊博士。兩位鼎力協助，審訂本書的內容，

讓學生能夠閱讀更多有趣的文章，增加他們學習統計的興趣。

如對本書有任何意見或建議，歡迎以郵寄、電話、傳真或電郵 方式聯絡教育局課程發展處數學教育組:

九龍油麻地彌敦道 405 號九龍政府合署 4 樓 教育局課程發展處

總課程發展主任(數學)收

(傳真: 3426 9265 電郵: ccdoma@edb.gov.hk) 教育局課程發展處

數學教育組

5

前言

香港統計學會一直致力向社會各界推廣對統計的認知。除了每 年與教育局合辦「中學生統計習作比賽」 （SPC） ，以鼓勵同學透 過團隊合作形式學習正確運用統計數據及增進對社會的認識 外，我們於 2009 年再與教育局合作創辦「中學生統計創意寫作 比賽」 （SCC） ，旨在鼓勵學生透過創意的手法，以及科學和客觀 的精神，用文字表達日常生活所應用的統計概念或利用統計概 念創作一個故事。

中學生統計創意寫作比賽已舉辦了十年，是時候停下來作出檢

討。回顧過去的參賽作品，我們看到同學對統計概念的認識深

入了，並能正確地運用統計知識作解說。得獎作品的整體質素

亦有所提升。本年度的比賽專題是「運動中的統計」 ，與同學的

生活較相關，同學很容易便能找到題材，創作故事。本屆參賽

作品約 70 份，數量與上屆相若。文章取材創新，趣味盎然；同

學活用各種統計知識創作故事，分析條理分明，解說清晰，值

得欣喜和嘉許。本書輯錄了本屆所有的得奬作品，藉此嘉許得

奬同學所付出的努力，並希望同學能夠從創作或閱讀這些得奬

作品中得到啟發，對統計知識有更深入認識。

6

我們藉此機會感謝籌備委員會和評審委員會全體成員對評審的 幫助和支持。他們的不遺餘力無疑是有助提高學生對統計的認 知和興趣。最後，感謝香港大學統計及精算學系贊助今屆比賽 的最佳專題寫作獎，和理大香港專上學院贊助今屆比賽的最佳 文章演繹獎。

籌委會主席 楊良河博士

總評審主任 張家俊博士

2019 年 11 月 27 日

7

目錄

編者的話 3 

前言 5 

目錄 7 

冠軍作品: 是否贏在起跑線才能贏到最後之名牌幼稚園的迷 思 ... 9 

亞軍作品: “Hotel, Really Trivago?” Discovering the Logic behind Hotel Selection of Trivago ... 26 

季軍作品: Safety Begins With Data ... 41 

優異作品 : 運動成績與體格的關聯 ... 59 

優異作品: 罰中有序 ... 68 

優異作品 : 一擊全中 ... 78 

優異作品: The crime journey of the three little pigs ... 92 

優異作品: NBA 勝率大謎團 主場客場逐個捉！ ... 109 

優異作品: 網絡資料審查員 ... 125 

優異作品 : 離婚率 ... 129 

邀請作品：淺談 NBA 統計 ... 134

邀請作品：大數據的應用與挑戰 ... 141

8

邀請作品：《標準差—何去何從？》 ... 147

邀請作品：Matrix Completion ... 155

9

冠軍作品:

是否贏在起跑線才能贏到最後之 名牌幼稚園的迷思

學校名稱：保良局何蔭棠中學 學生姓名：黃

穎璇

指導老師：陳智仁

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摘要：

「名校出狀元」等的報導，不斷鞏固了「要成功便要入讀名大學，要 入讀名大學便要入讀名中學，要入讀中學便要入讀名小學，要入讀名 小學便要入讀名幼稚園」的層層疊迷思，所以不少家長為了催谷年幼 的子女「贏在起跑線上」，不但爭相讓子女入讀名校幼稚園，更為他 們安排各種興趣班。然而國外多項研究卻發現，較遲入學的學生之學 術表現比早入學好，即「贏在起跑線」並無數據支持，甚或會令子女 有更多壓力。究竟這當中孰真孰假，入讀名牌幼稚園是否入讀名小學 的入場劵，又或者是以訛傳訛的美麗誤會呢？因此，本篇將會透過統 計和概率分析，探究是否要贏在起跑線入讀某某名牌幼稚園，才能贏 到最後。

今年文憑試的結果 出爐了，穎璇在社 交網站上看狀元究 竟花落誰家， 卻感 嘆地說：你們看，今 年的狀元又全是出 自傳統名校，果然 贏在起跑線這句話

是沒有說錯的，他們的家長從小為他們鋪路，讓他們入讀名牌幼稚園 和報讀各種的興趣班，讓他們能夠一條龍直升名牌小學和中學。假如 我們的父母以前有好好地讓我們贏在起跑線，也許狀元的名單上也可

11 能會有我們的一席

之位吧！

淑瑜卻以另一個網 頁的內容反駁說：

我才不認為要成功

必須要贏在起跑線，你看看大部分狀元也不是出自名牌幼稚園的，所 以我們相信也未必差過那些贏在起跑線的同學。正當二人在激烈討論 時，作為數學學霸的嘉俐開口了: 爭論是不會有結果的，我們何不去 計算分析？要計算兩者的關係，我們課堂上學過的概率、樹形圖等，

都是可以找出結論的。

名牌幼稚園名稱 直 升 其 小 學 的百分率

直升其直屬聯繫 中學的百分率

升讀大學百 分率 低主教幼稚園 55% 50% 35.9%

香港假光中學幼 稚園堅道

70% 72% 60%

香港假光中學幼 稚園

90%

聖減勒幼稚園 50% 49% 65%

聖保綠學校幼稚 園spk

80% 90% 71%

聖保綠學校幼兒 園spn

90%

12 名牌幼稚園名稱 直 升 其 小 學

的百分率

直升其直屬聯繫 中學的百分率

升讀大學百 分率 紋身書院幼稚園

部

77% 55% 74%

培根小學附屬幼 稚園

100% 80% 85%

協音小學附屬幼 稚園

75% 80% 69.7%

很快，穎璇就從網上找到了一大堆名牌幼稚園升學率的資科，但由於 資料數量的龐大，正煩惱著如何處理。然後嘉俐就建議說：要量度統 計 大 量 的 數 據 ， 我 們 可 以 運 用 算 術 平 均 數 ， 即 是 數 學 堂 所 學 的 arithmetic mean

1 2 3 1

1

1

n

n i

n i

i i

x x x x x

x x

n n n





   

   

 

名 牌 幼 稚 園 名稱

直 升 其 小 學 的 百分率

直升其直屬聯繫 中學的百分率

升 讀 大 學 百分率 平均率 76.3% 68% 65.8%

(註：假設每間名牌幼稚園的學生人數相等)

13

研究模型(1a)名牌幼稚園學生升大學的概率計算 (假設他們會由名牌幼稚園直升直屬小學，再直升到直屬中學)

名牌幼稚園學生升大學的概率 76.3% x 68% x 65.8% = 34.14%

名牌幼稚園 直屬小學 直屬中學

14

研究模型(1b)普通幼稚園學生升大學的概率計算 (假設他們由普通幼稚園升至普通小學再升到 Band1 中學)

階段 升讀百分率 解釋

幼稚園升小學 100% 全港的津貼小學，均以「小一

入學統籌辦法」收生，「統一 派位」會用隨機編號來分配 學位，所以每個人入普通津 貼小學的機率都是相同的。

小學升Band1中學 1/3  33.3% 根據香港津貼中學的「中學

學位分配辦法」，主要根據小

六 生 自 己 的Banding 及 選 校 意願，加上隨機編號來分配 中學學位，每個組別佔全港 學生的三分之一。

Band1中學升大學 54.78%

普通幼稚園學生升 大學的概率

100% × 33.3% × 54.78% = 18.24%

15

16

條件概率（英語：conditional probability）就是事件A在另外一個事件 B已經發生條件下的發生概率。條件概率表示為^{P A B}

 

B條件下A的概率」。

設 A 與 B 為樣本空間 Ω 中的兩個事件，其中

^{P B}   ^{ 0}

事件 B 發生的條件下，事件 A 發生的條件概率為：

 

^ _{ }

^

P A B

P B

 

在概率論中，樹形圖（Tree Diagram）是用來表示一個概率空間。

17

研究模型(2) 名牌幼稚園和普通幼稚園學生升大學的樹形圖 Band 1 中學 Band 2 中學 Band 3 中學

小學升中學的概率 _{33.3% 33.3% 33.3%}

中學升大學的概率 54.78% 17.44% 5.424%

普通小學學生升大 學的概率

33.3%54.78%

= 18.24%

33.3%17.44%

= 5.81%

33.3%5.424%

= 1.81%

名牌幼稚園 普通幼稚園

直屬小學

大學

普通小學

Band1 中學

Band2 中學

Band3 中學

大學

大學

大學

18

研究模型(2a) 名牌幼稚園學生升大學的概率計算

直屬小學 普通小學

名牌幼稚園升小學的概率 76.3% 23.7%

小學學生升大學的概率 68%  65.8%

= 44.74%

18.24%+5.81%+1.81%

= 25.86%

名牌幼稚園經不同中學升大學 的概率

76.3%  44.74%

= 34.14%

23.7%  25.86%

= 6.13%

名牌幼稚園學生升大學的概率 34.14% + 6.13% = 40.27%

研究模型(2b) 普通幼稚園學生升大學的概率計算

(

19

研究模型(2b) 普通幼稚園學生升大學的概率計算

直屬小學 普通小學

普通幼稚園升小學的概率 8%+2%=10% 100%  10%=90%

小學升大學的概率 68%65.8%

=44.74%

18.24%+5.81%+1.81%

= 25.86%

幼稚園經不同小學升大學 的概率

10%44.74%

=4.47%

90%25.86%

=23.27%

普通幼稚園學生升大學的 概率

4.47%+23.27% =27.74%

20

21

22

要「贏在起跑線」，未必一定要入讀直資或私立小學。教育局制度下 不少名牌英文中學屬於津貼聯繫學校、一條龍、直屬學校。入讀這類 小學，就更易升讀有聯繫的中學。

小學類別 定義 幼稚園升小學率

一條龍中小學 一條龍學校，大意是指小學和中學結 合成「一條龍」，「龍」內的小六畢業 生毋須考試或參加升中派位，便可全 數直升「龍」內的中學。

直屬中小學 直屬制度是整個統一派位制度的一部

分。根據教育局制定的中學學位分配 辦法，直屬中學經校方扣除重讀生及 自行分配學位數目後，可最多保留餘

額85% 學位予其直屬小學的學生。

23

小學類別 定義 幼稚園升小學率

聯繫中小學 聯繫制度是整個統一派位的一部分。

根據教育局制定的中學學位分配辦 法，聯繫中學經校方扣除重讀生及自

行分配學位後，可最多保留餘額 25%

學位予其聯繫小學的學生。

研究(3) 對比不同小學的升大學率

小學類別 小學升中學率 中學升大學率 小學升大學率

一條龍中小學 100% 38.1% 100%  38.1%

=38.1%

直屬中小學 85% 41.0% 85%  41.0%

=34.85%

聯繫中小學 25% + 33.3%

= 58.3%

42.24% 58.3%  42.24%

= 24.6%

名牌中小學 44.74%

普通中小學 25.88%

24 結論

雖然研究結果最後顯示名牌小學的入大學率都是最高，但是其實一條 龍和直屬小學的入大學率都高於普通小學，甚至跟名牌小學相差無幾。

可見即使同學輸在起跑線，沒有入讀名牌幼稚園，接下來也能夠入讀 一條龍和直屬小學去提高自己的優勢。因此，家長根本不必為了讓子 女贏在起跑線而費盡心思，這不但辛苦了自己,甚至會為孩子帶來沉

重壓力，令他們失去快樂的童年。「看來這個社會是沒有起跑線的。」

三人異口同聲地說，大家都對研究結果十分滿意。

(字數：2066 字)

25 參考資料：

[1] 基測百分百 HKDSE 優良率

http://blog.qooza.hk/ephmpntcj?eid=25924612&bpage=35 [2] 有直屬小學的中學-升學天地

https://www.schooland.hk/ps/feeder [3]【DSE放榜】DSE 9狀元龍虎榜 https://topick.hket.com/article/2112573/

[4] 全港一條龍中小學名單

https://www.schooland.hk/ps/through-train [5] 有聯繫中學的小學

https://www.schooland.hk/ps/nominated [6] 龍媽自製龍校幼稚園升小攻略 https://topick.hket.com/article/1445702/

[7] 贏在轉跑綫？趙榮德：很多DSE狀元來自普通幼小 https://topick.hket.com/article/1780071/

[8] 平均數- 維基百科，自由的百科全書 – Wikipedia

https://zh.wikipedia.org/wiki/%E5%B9%B3%E5%9D%87%E6%95%B0 [9] 條件概率- 維基百科，自由的百科全書 – Wikipedia

https://zh.wikipedia.org/wiki/%E6%9D%A1%E4%BB%B6%E6%A6%8 2%E7%8E%87

[10] 樹形圖- 維基百科，自由的百科全書 –Wikipedia

https://zh.wikipedia.org/wiki/%E6%A8%B9%E5%BD%A2%E5%9C%96

26

亞軍作品:

“Hotel, Really Trivago?”

Discovering the Logic behind Hotel Selection of Trivago

School Name: HKUGA College

Name of Students: HUNG Lok Ching Emily, LAM Kwan Yat Kyla, NG Hoi Tsing Tina

Supervising Teacher: Mr Timothy Ng

Abstract

Does Trivago actually recommend the cheapest prices of hotel choice?

Did they fake the users? Did they live up to their claims? In this article, the writers are going to explore these questions using probability and statistics.

27

* Lily and Joe are watching TV at home. The TV was playing advertisements.*

TV : Hotel? Trivago.

Lily : Ugh, it’s this advertisement again. Actually, does Trivago really works?

Joe : Well, Trivago claims that it “Compares the prices of over 600,000 hotels from over 200different websites” and “ Makes it easy for you to find the ideal hotel for the best price”.

Lily : Wow. If the claim was true, then it will be very impressive and effective to use. Let's try using it for the trip later this year. Our parents asked us to find the hotel for them, didn’t them?

Lily : Oh right. Let’s experiment using Trivago.

Lily : Here I have listed the requirements for this trip:

 People: 2 Adults and 2 children (7 and 12 years old respectively)

 Dates: 10/8/2019 - 15/8/2019

 Destination: Tokyo, Japan

Joe : I have just searched for a great hotel on trivago! It’s called Arist Bna Studio Akihabara.

28 Take a look.

Lily : Great! I am going to have some more researches on this hotel.

Joe : Let me click into the link of Expedia given by trivago. It shows that the price is $2542 per night. Hey, Lily. I remember that you have an Expedia app. I want to try to compare the prices to see if there are any differences. Can you search for this hotel on Expedia app for me, please?

Lily : Sure.

29

Joe : Wait, the price of the hotel shown on the Expedia link that trivago provides is $2090 per night. Why is it different?

Lily : I have just searched on the Expedia app, and it shows the price is

$2223 per night. What happened?

30

Comparing table of the hotel deluxe room (4 people)

NETWORK Trivago Expedia link that

trivago provides

Expedia App

PRICE (S) 2542 2090 2223

Joe : Look at the table above I have listed, the price varies between different platforms.

Lily : Wow. That’s shocking. I think we should check the original price the hotel provides as well to compare.

Joe : Good idea. Let me do it now.

31

Comparing table of prices on different platforms

NETWORK Trivago Expedia link that trivago provide

Expedia App

Original Price Point from Hotel

PRICE (S) 2542 2090 2223 2445

The difference from the original price by the hotel (Corr. to 3 sig fig)

+3.97% -14.5% -9.1% /

Joe : I have reorganised the table and compared the price points with the original.

Lily : Surprisingly, refer to this table, Trivago is actually even more expensive than the original price from the hotel. What’s the point of using Trivago then?

Joe : Same thoughts here. I think we should dig deeper, I feel like may be Trivago is lying.

Lily : Here’s the way this investigation to work. We have to collect mass data from different hotels and from different locations. The choices for the hotels must be random so it won’t be biased and unfair. Then we can compare the price points and get a reasonable conclusion.

Joe : For the location selection, I think we can use cities that we normally visit.

Lily : You always have such creative ideas! Here is the list I found.

32 1. Tokyo, Japan

2. Seoul, Korea 3. Taipei, Taiwan

Joe : Time for me to work now. First of all, the platforms providing prices we are going to compare will be:

 The original price from the hotel

 Trivago

 Expedia

 Hotel.com

 Agoda

 Booking.com

 Trip.com

 Wington.travel

Joe : Then, to decide the hotels of different locations, I think we should do 10 hotels for each location, which will be 3 × 10 = 30 hotels needed.

Lily : Okay no more talking, we still have a lot of work to do, let’s start now.

Joe : Here are the summary of steps I did.

1) Firstly, randomly select 10 hotels from Trivago website from the three cities: Tokyo, Seoul and Taipei. Fixed criteria: Stay at 10/8/2019 - 15/8/2019 for 2 adults, no breakfast.

2) Secondly, list prices recommended by Trivago and that of major hotel booking sites (6 of them) shown in Trivago

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3) Thirdly, check prices from hotel official sites

4) Next, for hotels of Tokyo city selected, check the prices of major hotel booking sites directly from their sites and compare them with that in Trivago

5) Then, observe the relation of the prices recommended by Trivago and that of all prices available in Trivago

6) Lastly, observe the trends of the prices found

Lily : I appreciate the efforts you paid, now please present your findings!

Joe : Here it is! Take a look first!

34

35

Lily : Oh, remember what the teacher taught us in school? We can calculate the mean and the standard deviation of these data.

Formula of standard deviation Formula of mean

Joe : I agree. Since standard deviation means a quantity expressing by how much the members of a group differ from the mean value for the group, we can compare the prices in a mathematical way.

Lily : Let me calculate the standard score of Mystays Premier Omori by Trivago first. By calculation, I found that :

36 The mean is:

(1464+1464+1464+1465+1606+1455+1455) /7

=1481.86

For the standard deviation: 50.84 For the standard score of Trivago:

(Trivago’s Price – Mean) / Standard Deviation (1464 –1481.86) / 50.84

≈ – 0.3513

Joe : Let me do the rest of the calculation for all other hotels.

Lily : Then what does it actually mean when all the standard score of Trivago is negative? Is it a bad thing?

37

Joe : Well, remember standard score calculates how much the price is far away from the average price. So being a negative valued standard score, it means it is lower than average. That’s not a bad thing, but a good sign. You always want a cheaper deal, right?

Lily : But most of the scores are between 0 and -1. So Trivago’s prices are actually only a bit lower than other sites….Hang on! Some scores are even positive!

Joe: You are correct. Trivago’s recommended prices are most of the time not the lowest price. With references to the table above, I’ve counted the total number of times while Trivago recommended price is not the cheapest. Each city I did for 10 hotels and mostly about 50% of the recommended prices are not the cheapest.

Location Number of times trivago recommended price is not the cheapest Tokyo 6

Seoul 5 Taipei 5

Lily : But doesn’t Trivago claim to provide the cheapest prices to us? Why are the recommended prices not the cheapest? Then maybe Trivago is a scam!

Joe : Don’t just jump into the conclusion. Trivago claims to provide the best prices. That doesn’t necessarily mean the cheapest price, that’s a misunderstanding. Based on the official website of trivago, it says:

38

The ‘our recommendations’ feature is based on a dynamic algorithm that shows you a range of attractive and relevant offers we think you’re going to love. In the ‘top position’ we display in green the offer which our algorithm recommends as a great offer. Our algorithm takes into account a number of relevant factors, such as your search criteria (for example your location and stay dates), the offer’s price, and its general attractiveness – for example, the experience we think you’ll likely have on the displayed booking site. We also take into account the compensation booking sites provide us with when a user clicks on an offer.

Lily : Oh so the recommended prices are not necessarily based on only price but also based on the past experiences and compensations provided.

Oh now I understand!

Joe : Secondly, prices shown in Trivago for the major booking sites are different from that of individual site.

No. of times Reason

Expedia.com 2 Trivago referral discount Hotel.com 2 Trivago referral discount Agoda.com 5 Trivago referral discount Booking.com 1 Trivago referral discount Trip.com 2 Wrong info in Trivago site

Wingon Travel 0 /

Lily: Why is that happening?

Joe: Major websites like expedia.com gave Trivago referral discounts. So if you click through trivago to expedia, the prices for the hotels may be

39

discounted. Meaning it may be cheaper. As you can see by the table, I have counted the number of times those major booking websites have trivago referral discounts and resulting to cheaper prices.

Lily : Wow. Agoda had 5 times of incidents, that’s pretty common.

Joe : Thirdly, hotel direct rates are always more expensive than the major booking sites as you can see in the table.

Lily : Is there any exceptions? I have noticed one in Tokyo.

Joe : Sure there will be some exceptions. I have counted them below:

Location Number of times hotel direct rates are cheaper than major booking sites

Tokyo 1 Seoul 0 Taipei 1

Lily : This is less than I expected. According to the finding, we may not order directly from the hotel anymore.

Joe : And for my last finding, I have noticed some of the websites constantly provide the same prices for hotels. I have also collected some interesting information. Actually, price pattern of the major booking sites reveals their ownership status.

Expedia and Hotel Under same group and prices are always the same Agoda and Booking Under same group but prices are sometimes the same Trip and Wing on Under same group but prices are sometimes the same

40

Lily: Wow, I didn’t know that. It’s almost midnight now, I think we should conclude things and go to bed soon...

Joe: Yah sure.

Lily: So after we have summed up all the points, below is the conclusion after all the work we had done:

1) Ignore Trivago’s recommendation price.

2) Comparing to booking directly with hotel, high chance to have a more favorable price by booking via hotel booking sites.

3) Comparing to booking via hotel booking sites, high chance to have a more favorable price by going through Trivago, so enjoy the referral discount.

(Word count: 2480)

References:

[1] Trivago Advertisement (English Version):

https://www.youtube.com/watch?v=Zv9UbMFWxnM

[2]https://support.trivago.com/hc/zh-tw/articles/360016108153-trivago-

%E6%88%91%E5%80%91%E7%9A%84%E6%8E%A8%E8%96%A6-

%E6%98%AF%E5%A6%82%E4%BD%95%E6%8E%92%E5%BA%8F

41

季軍作品:

Safety Begins With Data

School Name: HKUGA College

Name of Students: Aggie Chow, Timothy Chau, Tiffany Lee Supervising Teacher: Mr. Michael Yip

Word count: 2415 (including title)

Prologue:

Humans are soft beans, at any moment something as mundane as a car can kill us. While the thought of getting into a traffic accident wouldn’t usually cross our minds, it’s actually one of the major causes of death in developing countries. Of course, it’s impossible and impractical for society to stop using cars. But in the spirit of road safety, we decided to do what we can.

By studying when and where traffic accidents are most likely to happen, we can take extra precautions. Armed with this knowledge, perhaps our odds of surviving will increase…

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A Story Based on a Real Event

Image reference: John Chan's Instagram post This is a post from John Chan’s

Instagram, a secondary three student.

He recently witnessed a car accident which caused him late for school. Here is a conversation between Sam and John, about the accident that happened this morning.

“Hey Sam! You know what? I think I almost died today,”

“You’re over exaggerating, what happened?!” Sam asked

“This morning a car accident happened right in front of me! I had never thought that I would witness an accident first hand! It’s a terrible experience! ” John cried.

“Oh no! Were you okay?” Sam asked.

“I'm fine, but it was a little shocking to see it happened before my eyes. I hope that’ll never happen to me, anymore,” said John “so like, a thought popped into my mind: how can I avoid getting into a car accident?”

“You can’t! Unless you never step out your door!” Sam said.

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Like the stubborn person he always was, John wanted to prove Sam wrong.

He thought: if I analysed statistics about when and where traffic accidents are most likely to occur, I can avoid those places and time periods so that my chances of getting involved in one will be much smaller.

So, let's start analysing!

WHEN are traffic accidents most likely to occur?

Thus, Sam and John started to discuss about which day and which time traffic accidents were most likely to occur in. They both had their own opinions.

“I think traffic accidents are most likely to occur on Sunday because most people would be free to spend their weekend outdoors to have fun," said John. “No way! In that sense, wouldn’t there be the largest amount of accidents on Friday? Since more people would want to go out right after a long week of work” said Sam.

“Let's move on from this topic first and discuss about which time period traffic accidents are most likely to occur in,” said John.

“Okay, may be there isn’t a clear trend on which day traffic accidents are most likely to occur, but there must be a time where they usually happen?”

asked Sam.

“I think it should be around 6 p.m. to 7 p.m., because that's the time that most people finish their work and get back home. More vehicles,

44 more car accidents, right?” said John.

“Wait, I don't think so. Shouldn't it be seven o'clock to eight o'clock in the morning? Both students and workers have to travel during that period so there should be more accidents," said Sam.

“Hold up, we’ll never come to a conclusion like this. Let’s just have the data be the judge. Want to have a bet?” said John.

“Bring it on!”

Number of accidents in 7 days a week

45

After discussing with Sam, John found some data from the Transport Department of Hong Kong and analysed them using the chart above, by their time period and days of week.

“After getting this data, I found that the peak time of accidents is 1500- 1559 during Sunday.” said John.

“So you mean that we should avoid driving at 3 o'clock to 4 o'clock during Sunday?” said Sam.

“May be, but if we do, we should be extra careful during that time period,”

said John.

“Let’s look at the other chart, it indicates that Saturdays had the highest number of accidents…” said Sam.

“1700-1759 and 1800-1859 share the same number of accidents on Saturday,” said John.

“Wait, I’m confused. So should I not go out on Saturday or 1500-1559 during Sunday?” asked Sam.

“Just be careful on both will be fine,” commenced John.

“How about the trend?” said John?

“The number of accidents happened between the time period 0000-0659 were the less among all the time periods, which makes sense,” said Sam.

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“The number of accidents increases drastically afterwards, except for Sunday and Saturday.” John said.

“The number of accidents starts to decrease in 0800-0859. During lunch hour, the number of accidents increases again. The amount of accidents fluctuated between 100 and 190 during 1100-1159 and starts to decrease from 1800-1859 until the end of the day.” Sam said.

“But why does the line representing Sundays and Saturdays have a big difference from the Monday to Friday one?” John said.

“Let’s separate them into two charts and see what’s going on.”

So Sam and John continued on their quest for knowledge, at the expense of neglecting the work that they were supposed to do.

47

On Mondays to Thursdays, Traffic accidents typically peak at eight to nine o’clock in the morning and at six to seven o’clock at night. In those time periods many people are travelling from home to work and vice versa, so there are more cars on the roads and thus more traffic accidents are likely to occur.

There’s a big difference between the number of accidents happening between 20:00 to 06:59 and between 07:00 to 19:59, when accidents are much more likely to happen.

However, out of these four days, the time period with the most accidents happening would be 15:00 to 15:59 on Thursday, Thursdays seem to have slightly more accidents in general from 07:00 to 19:59 too.

“I also spotted that the blue line which represents Mondays' data, is slightly different from the other 3 days,” said Sam.

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“This is probably because of the black Monday effect, which everyone were simply sleepy and don’t want to work. As a result, accidents are most likely to occur due to the low level of concentration,” John said, pretending to be an expert.

“According to the chart, I have found out that most of the accidents actually happened during 1500-1559 on these three days! Probably because most of the family events end at this time. ” John said.

“Interesting. Also, maybe since Friday is a working day, the number of accidents is typically higher than Sunday and Saturday during the time period of 0800-0859, which matches what we derived from the ‘Monday to Thursday’ chart!” exclaimed Sam.

49

“We can conclude from the chart that the trends of Sunday and Saturday are similar as well. And the trend of Friday follows the trend of Monday to Thursday.” said John.

“Oh, it was my dad driving this morning! May be the type of vehicle used is a factor too?” asked John.

“Let’s look at this graph!” said Sam.

“Wow, this is amazing…..”

Sam: “What? The number of medium and heavy good vehicles only consist 5.6% of all types of car involved in accidents?!”

50

John: “It’s still reasonable to see private cars in the first place.”

Sam: “Um, I think this may be because the drivers working in public transport are well trained enough and there are simply more private cars on the road!”

John: “Then judging by the graph, taking public transport is actually safer than driving my own private car!”

We hypothesize that one of the ways to minimize your risk of getting into a car accident is by taking public transport instead.

WHERE are traffic accidents most likely to occur?

“John, where did you see the accident?”, asked Sam.

“Cross Harbour Tunnel.” John answered.

“Oh well, I bet accidents always happen there, since so many cars pass through there every day,” Sam said.

“Probably, but car accidents always happen in Tseung Kwan O Tunnel as well.” John said.

“Well, it seems that our opinions are not the same again. Then in which district do you think that car accidents are most likely to occur?” Sam asked.

“Um... let me think, Kwun Tong?”, John answered.

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“I think it is Sha Tin, ” Sam said.

Their opinions differed once more, so they started to search for data to analyse.

Out of the 18 districts in Hong Kong, Yau Tsim Mong district has the highest amount of car accidents happening there overall.

The number of car accidents happening in Yuen Long district and Sha Tin district are also noticeably higher. Yau Tsim Mong also has the highest amount of ‘light’ car accidents, followed by Yuen Long and Sha Tin once more. However, the district that has the highest amount of serious accidents, which is much higher than that of other districts, is Yuen Long.

Other districts have more or less the same amount (except for the Islands,

52

which have much less). The amount of car accidents that involved deaths is constant across all districts.

“Hold on,” interrupted Sam, “Would some districts have a particularly higher amount of accidents because their area is bigger and the population density is higher?”

“Oh yeah… I didn’t think of that,” John admitted.

However, that may not always be the case. Here we have a star plot representing area, population density, and amount of road accidents each district has. The size of the sector is proportional to the respective value it represents. Large districts with low population density are expected to have fewer accidents, e.g. Islands, North and Sai Kung Districts. In contrast, Yau Tsim Mong and Kwun Tong districts have high accident rate while their size is small and the density is high. Yuen Long and Tai Po districts are an exception though. They are accident-prone despite its large area and low population density.

53

“Hmm… That’s still too many exceptions to call it a clear trend,” Sam said,

“May be districts are too broad of an area to research on...”

“Let’s try looking at data on tunnels then!” suggested John.

54

Note- Area of the tunnels are defined by the area dictated in the law, which include the interior and exterior of the tunnels.

The tunnel with the most accidents was the Cross Harbour Tunnel, not surprising at all, considering that it is the tunnel with the most traffic in Hong Kong. However, the amount of accidents in a tunnel doesn’t always have to do with its traffic conditions. Below is a scatter plot illustrating the relationship between the amount of accidents a tunnel has and the amount of cars that pass through it every day.

55

Typically the higher the amount of average daily vehicles, the more accidents a tunnel has. But again, there are multiple exceptions. We can’t really reach a concrete conclusion since there isn’t much difference between the amounts of accidents each tunnel has.

In conclusion, Yau Tsim Mong and Kwun Tong are especially dangerous districts to drive in, and the Cross-Harbour Tunnel requires attention as well. The busiest areas always have the highest amount of traffic accidents, so we should be especially vigilant when driving through busy areas like those. But the population density and the average amount of vehicles daily don’t have a consistent relationship with the amount of car accidents an area has, so in other areas road conditions are probably a bigger factor.

56

WHAT WEATHER CONDITIONS are Traffic Accidents more likely to occur in?

“I’m not sure how useful this information can be, but there is one thing I’m sure of. Traffic accidents are more likely to occur in the rain, right?” said Sam.

“But judging by what we learnt today, the answer to that might be surprising,” John challenged, “We should do some research before coming to conclusions.”

“No way, there’s no reason why wet roads wouldn’t be more dangerous than dry roads- It’s harder for cars to stop when the ground is slippery, that’s why accidents are more likely to occur in rainy days, everyone knows that!” Sam exclaimed.

“Then why don’t you see for yourself?” John said.

57

“What?! The probability of getting into an accident on a wet road is much lower than the chance of getting into an accident on a dry road?” exclaimed Sam, “But how can that be?”

“I don’t really understand either,” confessed John, “Logically speaking, it would be more dangerous to drive on a slippery road, wouldn’t it? Maybe it is because people are especially alert when driving in dangerous conditions and end up performing better, or maybe more people take the MTR instead on those days.”

“That makes sense,” said Sam, “I guess there are things that we just can’t know for sure until we do more research.”

******

In the end, Sam and John forgot about the research they spent their precious time on. But then again, even if they followed the information they gained from these graphs and altered their lifestyle to be incredibly inconvenient, it still wouldn’t be a foolproof plan. After all as Murphy’s Law goes -

“Whatever can go wrong will go wrong”, so as long as cars exist, so will traffic accidents. The statistics shown today are a bit of extra knowledge, but please don’t follow them exactly. Ultimately, the best way to avoid traffic accidents is simply to be careful and stay alert at all times.

CONCLUSION

Collecting data and designing graphs were quite foreign tasks to us. After finishing the problem, we concluded that there isn’t a definite way to avoid

58

car accidents since there is an infinite amount of factors that can affect our topic, such as the state of the driver, problems with cars, or even a planned murder etc. While there may be a trend, unpredictable exceptions can always occur, especially when there are so many factors that lead up to such events, so it is best that we stay vigilant at all times.

Yet, we still want to do the best we can do to avoid car accidents and help readers of this project to be a safe person by analysing data related to road accidents.

In conclusion, we found out that using public transport can lower our probability for getting involved in car accidents, that we should avoid busy areas and rush hours, and that the weather doesn’t necessarily indicate safer or more dangerous conditions.

Stay safe!

References:

[1] Data from the transport department:

https://www.td.gov.hk/tc/road_safety/road_traffic_accident_statistics/201 7/index.html

[2] The census and statistics department:

https://www.censtatd.gov.hk/hkstat/sub/sp150.jsp?productCode=FA1000 96

[3] Hong Kong Weather Observatory:

https://www.hko.gov.hk/cis/statistic/rf_1_e.htm

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優異作品:

運動成績與體格的關聯

學校名稱：梁文燕紀念中學（沙田）

學生姓名：方肇楠，方偉華，林均俊 指導教師：陳志文老師

摘要：

相信大家在中學生涯中都參加過陸運會，大家當時會否羨慕在頒獎 臺上的同學呢？而你們又有沒有想過他們是憑甚麽獲勝呢？是出色 的技術嗎；還是過人的體格？在本篇文章中我們會探討運動員的身 高體重對其表現的影響。

左方為謙謙，右方為林林

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在體育課上，體育老師問了我們一個問題：大家的身形究竟跟跑步 有什麽關係呢？林林認爲越高大的人就跑得越快。謙謙認爲體重 越重就跑得越慢。其實在醫學界有一個叫BMI 的指標，BMI 的設 計是一個用於公眾健康研究的統計工具。當需要知道肥胖是否為 某一疾病的致病原因時，可以把病人的身高及體重換算成 BMI， 再找出其數值及病發率是否有線性關連。BMI 指數可以通過以下 公式計算：

W 是重量（公斤），h 是高度（米）。

體重指標 類別

18.49 或以下 過輕 18.5 – 22.9 適中 23.0 – 24.9 過重 25.0 – 29.9 肥胖

30.0 或以上 極度肥胖

圖表(一)

於是，我們就連同體育科老師和數學科老師一起收集及分析了全校 學生的體適能數據，並集合了全港中學生的體適能數據，再加以分 析，我們得出了以下結果:

項目 林林 謙謙 本校同學平均水平 全港學生平均水平

一分鐘仰臥起坐(次) 30 10 20.3 33.07

坐地前伸(釐米) 40 9 23.8 34.26

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項目 林林 謙謙 本校同學平均水平 全港學生平均水平

掌上壓(次) 35 8 13.5 19.43

九分鐘跑(米) 1700 1350 1473.1 1538.77

身高(米) 1.73 1.6 1.692 1.6966

體重(千克) 75 63.7 59.8 59.26

圖表(二)

經過計算，林林的BMI 指數為 25.0，而謙謙的 BMI 指數為 24.8，

看上去林林的身形高大，但是BMI 指數卻和謙謙差不多。從圖表

(二)中可看到，他們在長跑項目中相差 350 米的距離，但短跑又如 何？

讓我們回顧2016 年裡約奧運會男子 100 米的首八名：

2016 年夏季奧林匹克運動會田徑 男子100 公尺比賽

身高（米） 體重（公斤） BMI 指數

尤塞恩·博爾特 1.95 86 22.6

賈斯汀·加特林 1.85 79 23

安德烈·德格拉塞 1.76 68 21.9

約安·布雷克 1.8 76 23.4

阿卡尼·西姆拜恩 1.74 67 22.1

本·約瑟夫·梅特 1.79 70 21.8

吉米·維科 1.86 83 23.9

特拉伊瑪·布魯梅爾 1.73 70 23.3

平均值 1.81 74.875 22.75

圖表(三)

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經過計算，他們BMI指數的總體標準差為0.7228，不高。而他們身高 和體重的總體標準差分別為6.892和6.7535。以上的數據顯示，雖然身 高和體重有不同的分別，但是只要你有一個適當的BMI指數，即是身 高和體重有一個一定的比例，你便能在短跑中取得佳績。我們在這裡 也可以作出一個大膽的假設：林林和謙謙的短跑時間不會相差太遠。

為了測試我們的假設是否正確，我們特別邀請了林林和謙謙進行一次 短跑100米競賽，林林的成績為13.75秒；而謙謙的成績是14.19秒，當 中只相差0.44秒，證明了林林和謙謙的BMI指數在短跑這項運動中表 現相仿。

除了短跑，其實其他運動都有一個特定的BMI指數範圍，該範圍內的 人取得勝利的機會很大。為了證明，我們也找到了馬拉松的世界紀錄 的前三名的BMI指數，分別為埃利烏德·基普喬蓋：20.0, 鄧尼斯·基普 魯托·基梅托：18.8，肯內尼薩‧貝克勒：21.0。可以看到馬拉松這類長 跑的選手BMI指數比較低，以上三位的BMI指數平均值為19.9。

謙謙和林林的 BMI 指數那麼高，有甚麼運動是適合他們的呢？2016

裡約奧運的鉛球冠軍里安·克勞澤身高 2.01 米，體重 132 公斤，BMI

指數32.1，但這對他們來說還算是太高了，而且我相信對香港的大部

分學童也不是十分適合的。此時我們的另一位朋友—小明走了過來參 與我們的討論，他是我們學校鉛球比賽的記錄保持者，他剛剛做完手 握力測試，於是他便問手握力的強度是否與推鉛球有關，爲了解答他 的疑問，我們再次收集了我校鉛球比賽八強的手握力數據作出比較：

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鉛球比賽八強的手握力 手握力數據 (千克) (左手+右手)

體重 (千克)

1. 79.5 85.0

2. 76.2 84.7

3. 75.6 76.8

4. 71.3 78.6

5. 71.0 65.9

6. 69.8 70.5

7. 68.7 69.0

8. 67.0 67.1

平均值 72.3875 74.7

圖表(四和五)

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從以上圖表可得知，發現手握力的強度與推鉛球的距離(成績)真的有 關聯。在圖表(四)中可看到第一名的手握力數據最高，有79.5千克，

然後手握力的強度隨著參加者的名次下降，到67千克: 有越高的手握 力能在推鉛球中取得更好的成績。

回到我們的主題，參加者的BMI 指數與手握力(及推鉛球成績)又是否

有關？我們根據推鉛球八強的體重及手握力數據繪畫了圖表(五)進 行比較。我們發現兩者成正比，參加者體重越高，手握力的指數便越 高。透過繪畫圖表後，我們便要尋找該條線性回歸的直線方程(最佳 配適線)，這次我們只需要進行簡單計算便能夠計算出圖中線性回歸 的直線方程。

A B C D E F

x 體重(kg) y 手握力數據(kg) ^૛

85 79.5 10.3 7.1125 73.25875 106.09

84.7 76.2 10 3.8125 38.125 100

76.8 75.6 2.1 3.2125 6.74625 4.41 78.6 71.3 3.9 -1.0875 -4.24125 15.21

65.9 71 8.8 -1.3875 12.21 77.44

70.5 69.8 -4.2 -2.5875 10.8675 17.64 69 68.7 -5.7 -3.6875 21.01875 32.49 67.1 67 -7.6 -5.3875 40.945 57.76 體重的

平均數

手握力的平均 數

總和 總和

74.7 72.3875 198.93 411.04

鈄率(m) 0.483967 y 軸截距(c) 36.23513

圖表(六)

在圖表(六)中，我們進行運算。該條直線方程的斜率只需將E欄總和除 以F欄總和，便能得出0.484(斜率, m) 。而y軸相交點(c)則為36.235。

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圖中線性回歸的直線方程是 y 0.484x36.235。十項全能的冠軍 羅曼·謝布爾勒的身高體重為1.86米，體重為88公斤，BMI指數為 25.8，對他們來說非常適合。

在最後，如果你希望在運動中取得佳績，根據上述的研究，我們可以

得出運動員的成績是與 BMI 指數直接掛鈎的。就如上述，謙謙和林

林的 BMI 指數與奧運選手大致相約，因此，推論他們能獲得獎項，

而事實證明他們在學校陸運會中取得獎項。而在鉛球比賽中，小明能 取得佳績亦是和體重與手握力息息相關。體重和手握力越高，在比賽 中取得的名次越高。因此，我們可以證明我們的推論是正確的。根據 以上對鉛球和短跑的分析。我們可以大膽地假設運動成績是和身體質 量指數成正比。我們根據這個猜想，搜集了上年度陸運會鉛球的冠軍 和他的各項身體數據，見以下圖表(七):

圖表(七)

在名次一欄中，第一名的推鉛球紀錄是8.84米，再參考圖表(八)看看 他的體重和身高，發現其BMI指數和手握力數據和紀錄保持者的相 似，因此他在比賽中取得佳績。由此我們發現身體質量指數能幫助

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我們在相應的運動項目中取得優勢，較易獲取好成績。

記錄保持者 上任鉛球冠軍

體重 90.5 千克 95.1 千克

身高 170 厘米 180.5 厘米

BMI 指數 31.3 29.1

手握力 79 公斤（左手加右手） 76.5 公斤（左手加右手）

圖表(八)

因此，各位在選擇運動時可以參考一些運動員的身體質量指數來選擇 一個適合自己的運動了。

67 參考資料：

[1] 維基百科: 2016年夏季奧林匹克運動會田徑男子100公尺比賽 [2] 梁文燕紀念中學（沙田）二零一八至二零一九陸運會數據 [3] 梁文燕紀念中學（沙田）體育科體適能數據

[4] 香港中學生體適能常模表：

https://cd1.edb.hkedcity.net/cd/pe/tc/rr/pfs/sec_09_10_c.pdf

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優異作品:

罰中有序

學校名稱：中華傳道會安柱中學 學生姓名：陳澤聲、曾智翹、伍俊聲

指導教師：周恩隆老師

摘要：

NBA(National Basketball Association) –國家籃球協會，是不少籃球粉 絲所喜愛和關注的熱門話題。當俊聲看到有關 James Harden 在主場 的傑出罰球表現的新聞時，立即與自己的好朋友智翹和澤聲討論有關

主場優勢的問題。他們嘗試運用統計學作分析，以NBA 球星的罰球

數據來尋找出箇中的「主場之利」。

69 一天，俊聲、澤聲和智翹看到以下新聞:

「香港時間3 月 1 日，火箭在主場以 121-118 險勝熱火，本場比賽，占士‧夏登出戰 44 分鐘32 投 16 中，三分球 18 中 8，罰球 18 中 18，轟下 58 分 7 籃板 10 助攻 4 偷球 1 封籃。這是夏登生涯第 59 次單場至少罰進 15 球，高居 NBA 歷史第一。」

(新聞內容節錄)

俊聲：哇，James Harden又入這麼多罰球了！依我看，一定是因為

「主場優勢」的緣故。

澤聲：主場優勢？甚麼是主場優勢呢？

俊聲：主場優勢，顧名思義，就是因為比賽在主場（自己球隊的體育 館）舉行，所以球員在環境和心理上有優勢，例如更熟悉比賽場地及 當地的氣候，以及由於支持者佔大多數，因而能夠得到更多的加油助 威，並對對手施加更大的壓力。

澤聲：原來如此！這麼看來，James Harden 的罰球表現很可能就是 因為主場才能這麼出類拔萃！

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俊聲：要證實James Harden 的罰球表現是不是在主場較好，不如我 們來統計一下？

澤聲和智翹：好！

俊聲：今天是2019年3月9日，那麼我們就只統計由開季(2018年10月 17日)至今的賽事吧。同時，我們可以來主場和客場的罰球表現作對 比。

智翹：截至2019 年 3 月 9 日，James Harden 的罰球表現如下：

表一、2018-2019 James Harden 主客場罰球表現 主場

罰球次數 主場 罰球進球數

主場 罰球命中率

客場 罰球次數

客場 罰球進球數

客場 罰球命中率 James

Harden 375 336 89.6% 312 267 85.6%

智翹：我們可以計算罰球命中率來比較和清楚表達主客場的罰球表現。

罰球命中率公式

罰球命中率 罰球進球數

罰球次數

當主場或者客場罰球命中率越高，代表球員在主場或者客場罰球表現 較好；反之，當主場或者客場罰球命中率越低，則代表球員在主場或 者客場罰球表現較差。從中可見，James Harden 的主場罰球表現明顯

比客場較佳，主場罰球命中率比客場罰球命中率約高4%。

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澤聲：看來 James Harden 確實擅長於主場。這麼說，球員在主場，

罰球的表現是不是就較好呢？

智翹：也不能一概而論，不同球員在不同的地點有不同的能力，例 如以下的一組球員，其中五人（除了Lebron James）皆擔任後衛，

而Lebron James, James Harden 和 Stephen Curry更曾獲得最有價值 球員(MVP，Most Valuable Player)的殊榮，對各自球隊有著豐功偉 績：

表二、2018-2019 球員主客埸罰球表現

主場罰球 次數

主場罰球 進球數

主場罰球 命中率

客場罰球 次數

客場罰球 進球數

客場罰球 命中率

命中率差 (主—客) James Harden 375 336 89.6% 312 267 85.6% 4%

Lebron James 183 119 65.0% 178 120 67.4% -2.4%

Klay Thompson 83 66 79.5% 57 48 84.2% -4.7%

Stephen Curry 107 96 89.7% 132 123 93.2% -3.5%

Kyrie Irving 92 78 84.8% 98 87 88.8% -4%

Damian Lillard 191 173 90.6% 233 213 91.4% -0.8%

平均值 172 145 84.2% 168 143 85.0% -0.8%

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主場罰球命中率(%) 客場罰球命中率(%)

智翹：以棒形圖來看：

2018-2019 球員主客場罰球表現

（棒形圖：球員主客場罰球表現）

智翹：其中，平均值(Mean)是指統計對象的一般水平，也是描述數 據集中趨勢的一種方法。我們既可以用它來反映一組數據的一般情 況，也可以用它進行不同組數據的比較，以看出組與組之間的差 別。要計算平均值，可以利用以下的公式。

平均值公式

(n =數據的數量)

智翹：在這裡，我們就用平均值比較了六位球員主客場的罰球表

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現。客場罰球表現較佳，儘管受到James Harden 這個極端值影響，

客場罰球命中率比主場罰球命中率仍然高了0.8%

澤聲：極端值？

智翹：極端值（Extreme values）是指在統計中，特別大或特別小的數 值。在這裏James Harden 的主場和客場發球次數為大約 300 以上，明

顯比其他球員的100 左右高出很多，是一個極端值。極端值對於數據

的平均值影響很大，因此這時候我們可以利用更多球員的數據來減少 極端值的影響，例如在我們這裏就利用了六個球員的數據。

澤聲：你真是人如其名的「智多星」！那麼表中出現的命中率差又是 甚麼呢？

智翹：命中率差，顧名思義，是主場罰球命中率和客場罰球命中率的 差異（主場罰球命中率-客場罰球命中率），用以反映主場和客場的罰 球表現差距。當命中率差為正數，代表主場罰球表現比客場較為良好；

若為負數，則代表客場罰球表現比主場較為良好。六個球員中，五個 球員的命中率差都為負數，反映這些球員中，客場罰球表現普遍較好。

澤聲：命中率差和平均值反映這一組球員的客場罰球表現的都優於主 場，看來真不可以一概而論。

俊聲：要知道球員的罰球表現無法單憑一個賽季來下定論，不如我們 再統計一下James harden過往的表現？

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智翹：好，經過搜集網上資料統計，以下是James Harden 過往十年，

即2009-2010 賽季至 2018-2019 賽季的主客場罰球表現：

表三、過往十年James Harden 主客場罰球表現

年份 賽季 主場罰

球次數

主場罰球 進球數

主場罰球 命中率

客場罰 球次數

客場罰球 進球數

客場罰球 命中率

命中率差 (主—客) 1 2009-2010 131 100 76.3% 109 94 86.2% -9.9%

2 2010-2011 157 139 88.5% 186 150 80.6% 7.9%

3 2011-2012 181 156 86.2% 188 156 83.0% 3.2%

4 2012-2013 398 338 84.9% 394 336 85.3% -0.4%

5 2013-2014 330 291 88.2% 335 285 85.1% 3.1%

6 2014-2015 408 350 85.8% 416 365 87.7% -1.9%

7 2015-2016 406 356 87.7% 431 364 84.8% 2.9%

8 2016-2017 466 390 83.7% 415 356 85.8% -2.1%

9 2017-2018 419 352 84.0% 308 272 88.3% -4.3%

10 2018-2019 375 336 89.6% 312 267 85.6% 4%

總數 3271 2808 85.8% 3094 2645 85.5% 0.3%

平均值 327.1 280.8 85.8% 309.4 264.5 85.5% 0.3%

智翹：也可以參考下圖James Harden 在過往十年的罰球表現：

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智翹：從資料可見，James Harden在過往十年的罰球表現中，主場表 現稍微佔優：平均值方面，主場罰球命中率的平均值為85.8%，比客 場罰球命中率85.5%高了0.3%；命中率差方面，主場表現仍然較好，

中位數為1.25%，反映主場罰球表現較好。

俊聲：甚麼是中位數?

智翹：中位數（median）也是描述數據集中趨勢的一種方法，它是 一組數據中位於中間位置的數字，因此不容易被極端值影響。計算 中位數時要由小至大排列。要計算中位數，也可以利用以下兩種公 式，分為偶數和奇數：

中位數公式

第 項 若 為奇數

第 項 第 項 若 為偶數

智翹：計算命中率差時，上表中有十個數據，屬於偶數，因此中位數 是第五（-0.4%）和第六個數據（2.9%）的總和除以二，得出命中率 差中位數為1.25%。

澤聲：主場罰球命中率在平均值和中位數上都比客場要高，可以看到 主場優勢確實存在。同時以最大值（maximum）來看，主場罰球命中

率曾經到達本年度 89.5%的歷史高位。而客場罰球命中率只到達

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87.7%，比主場少了 1.8%，反映 James Harden 在主場罰球的潛力比客 場要高。

智翹：沒錯，當我們比較球員主客場的表現時，除了考慮整體表 現，也要考慮到球員在主客場的潛力，這個時候我們就可以用最 大值比較。

俊聲：既然James Harden在主場的潛力比客場大，這是不是說明 主場優勢較大？

智翹：是的，不過現在有些球員更加熟悉其他環境，主場優勢並不 那麽顯著。正如表二部分球員在客場的罰球表現意料之外地比主場 還要好。

澤聲：不過還是有球員，在主場優勢影響下，像 James Harden一樣在 主場發揮較好。

俊聲：（不勝其煩地）你們慢慢討論，我先打一會籃球……

智翹和澤聲：豈有此理，等等我們啊！

(字數：2374 字)