Statistics-Chapter 3: Central Tendency

Abdullah Al Mahmud

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Chapter 03 (Central Tendency)

What is Central Tendency?

Why needed?

                   mpg cyl  disp  hp drat
Mazda RX4         21.0   6 160.0 110 3.90
Mazda RX4 Wag     21.0   6 160.0 110 3.90
Datsun 710        22.8   4 108.0  93 3.85
Hornet 4 Drive    21.4   6 258.0 110 3.08
Hornet Sportabout 18.7   8 360.0 175 3.15
Valiant           18.1   6 225.0 105 2.76
Duster 360        14.3   8 360.0 245 3.21
Merc 240D         24.4   4 146.7  62 3.69
      mpg             cyl           disp      
 Min.   :14.30   Min.   :4.0   Min.   :108.0  
 1st Qu.:18.55   1st Qu.:5.5   1st Qu.:156.7  
 Median :21.00   Median :6.0   Median :192.5  
 Mean   :20.21   Mean   :6.0   Mean   :222.2  
 3rd Qu.:21.75   3rd Qu.:6.5   3rd Qu.:283.5  
 Max.   :24.40   Max.   :8.0   Max.   :360.0
  • Summary
  • Comparison
  • A value to represent all

Criteria for a Good Measure of Central Tendency

  • Well-defined
  • Understandable
  • Considers all values
  • Suitable for further analysis
  • Not affected by sample fluctuation

Measures (Averages)

  • Arithmetic Mean (AM)
  • Geometric Mean (GM)
  • Harmonic Mean (HM)
  • Median
  • Mode
  • Partition Values (Quartiles, Deciles, Percentiles etc.)

AM

\(AM=\bar x=\frac{\sum x}{n}\)

If there are frequencies or weights

\(\bar x=\frac{\sum f_i x_i}{\sum f_i} \space or \space \frac{\sum w_i x_i}{w_i=n}\)

Find AM

 [1] 10 55 38 48 51 25 14 44 23 22
Answer

Find AM: 2, 2, 3, 3, 5, 5, 5, 8, 8, 9

There are 2 ways.

\(\bar x = \frac{2+2+...+9}{10}\)
Answer

Mean Using Frequency

For grouped data

Working hours
(x)		 Employee
(f)		 fx
2 2 4
3 2 6
5 3 15
8 2 10
9 1 9
\(\sum f =10\) \(\sum fx = 50\)

\(\therefore \bar x = \frac{\sum fx}{\sum f} =\frac{50}{10}=5\)

Freuency vs Weight

Suppose, different judges give different scores, but not all evaluation has same weight.

Judge Rating
(x)		 Weight
(w)		 wx
1 8 2 16
2 7 3 21
3 4 5 20
4 5 1 5
5 7 3 21
\(\sum w_i = 14\) \(\sum w_ix_i = 83\)

\(\therefore \bar x = \frac{\sum w_ix_i}{\sum w_i}\)

Answer

Shortcut Method for AM

Calculate the mean in a smart way

[1] 1009 1037 1047 1024 1013 1043
Show

Shortcut Method Formula

Consider the values: 1005, 1010, 1015

If 1000 is subtracted: 5, 10, 15

If again divided by 5: 1, 2, 3

Converted Mean = 2

Original Mean = \(2 \times 5 + 1000=1010\)

Show

Properties of AM

Also look at this

  • \(\sum (x_i-\bar x)=0\); can you prove it?
  • \(\sum (x_i-\bar x)^2 \le \sum (x_i-a)^2, \space a \ne \bar x\)
  • Depends on change of origin and scale?
  • \(\bar x + \bar y =\frac{\sum x+\sum y}{n_x+n_y}\)
  • Combined mean: \(\bar x_c=\frac{n_1 \bar x_1+n_2 \bar x_2+...+n_k \bar x_k}{n_1+n_2+...+n_k}\)
  • \(AM\ge GM \ge HM\) & \(AM \times HM = (GM)^2\)
  • AM of first n natural numbers = \(\frac{n+1}{2}\)

Arithmetic Mean of Series

\(11,13,15, \cdots, 57\)

Tip: Let \(c=2nd \space Term - 1st \space Term\) and \(a=1st \space Term -c\)

\(U = \frac{X-a}{c}\)

(Dis)advantages of AM

  • Well-defined
  • Less affected by sample fluctuation
  • Comparison among sets is easy
  • Uses all values
  • Suitable for further analysis.
  • Affected by outliers

Geometric Mean (GM)

\(GM=(x_1 \times x_2 \times \cdots \times x_n)^{(1/n)}\) or \(GM=(x_1^{f_1} \times x_2^{f_2} \times \cdots \times x_n^{f_n})^{(1/\sum f_i)}\)

Find GM: 2, 4, 6

Answer

Try this one: 20020, 30080, 50086, 40130

Concept of Logarithm

“An admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.”

— Pierre-Simon Laplace

  • Log and Antilog
  • \(log_24=?\)
  • if \(log_2x=3\), then x=?

GM using Log

  • \(GM=(x_1 \times x_2 \times \cdots \times x_n)^{(1/n)}\)
  • \(log (GM) = \frac 1n \{log(x_1)+log(x_2) + \cdots + log(x_n)\} = A (suppose)\)
  • \(\therefore GM = Antilog (A)\)

GM Easier Formula

x = 20020, 30080, 50086, 40130

  • log x = 9.9, 10.31, 10.82, 10.6
  • Mean of logx = \(\frac{\sum logx}{n}\)
  • Original Mean = \(antilog(\frac{\sum logx}{n})\)
Answer

Calculate GM

Marks # Students
10-12 4
12-14 5
14-16 3
16-18 5
18-20 7
20-22 2

Make a table using these columns: \(x_i, f_i, logx_i, f_ilogx_i\)

  • \(GM =antilog(\frac{\sum f_i logx_i}{\sum f_i})\)
Answer

(Dis)advantages of GM

  • x = 5, 10, 15, 20, 100, 1000

    log(x) = 0.7, 1, 1.18, 1.3, 2, 3

Velocity Paradox

Story of oil scam

S = 150 km

\(v_1=\space 10 km/h, v_2=\space 15 km/h, v_3=\space 20 km/h\)

What is the average speed?

  • \(AM=\frac{10+15+20}{3}=15 \space km/h\)
  • Think more fundamentally
  • \(\sum S=3\times 150=450\)
  • \(t_1=15h, t_2=10h, t_3=7.5h\)
  • \(\bar v = \frac{\sum S}{\sum t}=\frac{450}{32.5}=13.84 \lt AM\)
  • \(=\frac{450}{15+10+7.5}\)
  • \(=\frac{450}{\frac{150}{10}+\frac{150}{15}+\frac{150}{20}}\)
  • \(=\frac{450}{150(\frac{1}{10}+\frac{1}{15}+\frac{1}{20})}\)
  • \(=\frac{3}{\frac{1}{10}+\frac{1}{15}+\frac{1}{20}}\)
  • \(=\frac{3}{\frac{1}{v_1}+\frac{1}{v_2}+\frac{1}{v_3}}\)

Harmonic Mean

Formula: Reciprocal of Mean of \(\frac{1}{x_i}\)

Reciprocal of \(\frac{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}{n}\)

Thus, \(HM = \frac{n}{\sum \frac{1}{x_i}}\)

Calculate: 2, 4, 8

Answer

For grouped data

  • \(HM=\frac{\sum f}{\sum \frac{f}{x}}\)
  • For wighted data: \(\frac{\sum w}{\sum \frac{w}{x}}\)

Why and When HM

  • When there are rates associated, say speed, and numerator is fixed.
  • \(Speed, v = \frac{S}{t}\); HM if S is fixed
  • Example: A man travels 120 km the first day at 12 kph, the same distance at 10 kph on the 2nd day, and at 8 kph on the 3rd day. Find his average speed.

Wighted AM vs Weighted HM

Suppose, a bus travels 10 km at 10 kph, another 15 km at 20 kph, and another 20 km at 25 kph. What is the average speed.

HM \(\rightarrow\) consider distances as weights

AM \(\rightarrow\) consider times as weights

Time, \(t=\frac d v=\) 1, 0.75, 0.8

  • \(WHM = \frac{10+15+20}{\frac{10}{10}+\frac{15}{20}+\frac{20}{25}}\)=17.65
  • \(WAM = \frac{1\times 10 + 0.75 \times 20 + 0.80 \times 25}{1+0.75+0.80}\) = 17.65
  • True mean, \(\bar v=\frac{\sum d}{\sum t}=\frac{45}{2.55}=\) 17.65

HM Example 2

  • A passerby travels 10 km at 20 kph, 5 km at 15 kph, and 4 km at 12 kph. What is the average speed? (Use weighted HM)

  • Here, distances are different. Consider them weights.

  • \(HM=\frac{w_1+w_2+w_3}{\frac{w_1}{x_1}+\frac{w_2}{x_2}+\frac{w_3}{x_3}}\)
  • HM = 16.286

Median

Find Median and AM

1,3,5,6,7,8

1,3,5,6,7,8,120

  • Median is a better measure
  • What now: 10,12,15,15,20, 50, 70,75,80

Quadratic Mean

\(QM=\sqrt{\frac{x_1^2+x_2^2+...+x_n^2}{n}}=\sqrt{\frac{\sum x_i^2}{n}}\)

  • Also known as Root mean square
  • More at: https://en.wikipedia.org/wiki/Root_mean_square#Uses
  • For grouped data?

Partition Values

Find medians

x = 4,5,6,8,9,11,16

y = 4,5,6,8,9,11,16,19

  • Median of x is 8
  • Median of y is 8.5
  • Formula for odd n = \(\frac{n+1}{2}th \space value\)
  • Formula for even n = \(\frac{\frac{n}{2}th+(\frac{n}{2}+1)th}{2}\)

Dis(advantages) of Median

  • Unaffected by outliers (extreme values)
  • Graphically estimable
  • Affected by sample fluctuation
  • Not based on all values
  • Not suitable for further mathematical analysis

Quartiles, Deciles, and Percentiles

\(Q_1= \frac{(n+1)}{4}th \space\)

\(Q_2, Q_3=?\)

Find \(Q_1, Q_2, Q_3\)

X = 4, 6, 7, 10, 12, 13, 14, 15, 16, 17, 19

See more partition names

  • 4, 8.5, 13, 12.0909091, 15.5, 19
  • What are the formulae for Deciles (\(D_i\) and \(P_i\))

General Formula for Partition Values

For odd n,

\(A_i= \frac{i \times (n+1)}{k}th \space value\)

For even n,

\(A_i=\frac{\frac{i \times n}{k}th+(\frac{i\times n}{k}+1)th}{2}\)

where, k = no. of partitions

For median, for example, k = 2.

  • What if we divide the data set into 20 segments?

Find Deciles and Percentiles

X = 4, 6, 7, 10, 12, 13, 14, 15, 16, 17, 19

Find \(D_3, D_8, P_{17}, P_{56}, P_{93}\)

Averages from Grouped Data

Age # Employees CF
20-24 40
25-29 60
30-34 200
35-39 180
40-44 150
45-49 110
50-54 175
55-59 60
60-64 25

Median: \(Me = L + \frac{\frac{n}{2}-F_c}{f_m}\times c\)

Mode: \(M_o=L+\frac{\Delta_1}{\Delta_1+\Delta_2}\times c\)

Quartiles: \(Q_i = L + \frac{\frac{in}{4}-F_c}{f}\times c\)

Find AM, Me, Mo, Q, and \(P_{30}\)

Partition Values from Graph

  • Draw an Ogive
  • Draw a straight line parallel to X-axis at the point \(\frac n 2\) of Y-axis. - Draw a perpendicular on X-axis from the intersection point.

Graph

Comparison of Averages

Property AM GM HM Median Mode
Formula Easy to
understand
Easy
Easy
Easy
Easy
Considers
values		 All All All Middle term(s) Highest frequency
Computaion Easy
Easy
Easy
 Easy Easy
Effect of
Outliers		 Highly affected Less affected
than AM		 Less affected
than GM		 Unaffected Can be highly
affected
Effect of Sampling
Flcutuation		 Less affected Less affected Less affected Can be highly
affected		 Can be highly
affected
Suitability for
further analysis		 Possible Possible Possible
Possible
Possible

When AM = GM = HM

If \(x_1=x_2=x_3\); prove it

Theorems

Sum (x - mean)

\[\sum_{i=0}^n (x_i-\bar x)=0\]

\[\sum_{i=0}^n f_i(x_i-\bar x)=0\]

Square of deviation from mean

\[\sum_{i=1}^n (x_i-\bar x)^2 \lt \sum_{i=1}^n (x_i-a)^2; a\ne\bar x\]

\[\sum_{i=1}^n f_i(x_i-\bar x)^2 \lt \sum_{i=1}^n f_i(x_i-a)^2; a\ne\bar x\]

AM - Change of origin & scale

Prove that AM depends on origin and scale Use frequency as well i.e,

  • \[\bar x=\frac {\sum_{i=1}^nx_i}{n}\]
  • Let \(u_i = \frac{x_i-a}{c}\)
  • \(x_i = cu_i + a\)
  • \(\sum x_i =\)
  • \(\bar x = c \bar u + a\)

AM ≥ GM ≥ HM

  • \((a-b)^2 \ge 0\)
  • \((a+b)^2 - 4ab \ge 0\)
  • \((\frac{1}{a}-\frac{1}{b})^2 \ge 0\)

Full Proof

We know, \[\begin{eqnarray} & &(a-b)^2\ge 0 \nonumber \\ & \Rightarrow & (a+b)^2-4ab \ge 0 \nonumber \\ & \Rightarrow & (a+b)^2 \ge 4ab \nonumber \\ & \Rightarrow & (a+b) \ge 2 \sqrt{ab} \nonumber \\ & \Rightarrow & \frac{a+b} 2 \ge \sqrt{ab} \nonumber \\ & \Rightarrow & AM \ge GM \nonumber \\ \end{eqnarray}\]

Similarly, \[\begin{eqnarray} & &(\frac{1}{a}-\frac{1}{b})^2\ge 0 \nonumber \\ & \Rightarrow & (\frac{1}{a}+\frac{1}{b})^2 -4 \cdot \frac 1 a \cdot \frac 1 b\ge 0 \nonumber \\ & \Rightarrow & (\frac{1}{a}+\frac{1}{b})^2\ge \frac 4 {ab} \nonumber \\ & \Rightarrow & (\frac{1}{a}+\frac{1}{b}) \ge \frac 2 {\sqrt{ab}} \nonumber \\ & \Rightarrow & \sqrt{ab}(\frac{1}{a}+\frac{1}{b}) \ge 2 \nonumber \\ & \Rightarrow & \sqrt{ab} \ge \frac{2}{(\frac{1}{a}+\frac{1}{b})} \nonumber \\ & \Rightarrow & GM \ge HM \nonumber \\ \end{eqnarray}\]

AM × HM = GM²

Prove using formulae

For two non-zero positive numbers, \(AM \times HM =(GM)^2\)

Mean-Median of Natural Numbers

Mean and Median of first n natural numbers are \(\frac {n+1} 2\)

Combined mean

Form A B
# Cadets 24 25
Mean Score 75 70

What’s the combined mean?

  • \[\bar x_c=\frac{n_1 \bar x_1+n_2 \bar x_2}{n_1+n_2}\]

Mean of U = X + Y

If \(u=x+y, \bar u=\bar x + \bar y\); if \(n_1=n_2=n\)

Given \(u=x+y\)

  • \(\bar u = \frac{\sum u}{n}\)
  • \(\frac{\sum (x+y)}{n}\)
  • \(\frac{\sum x}{n}+ \frac{\sum y}{n}\)
  • \(\bar x + \bar y\)

GM of two variables

For equal number of observations, GM of two variables is equal to the product of their individual means.

  • \(G_1 = (x_1 x_2 x_3)^{1/3}\)
  • \(G_2 = (y_1 y_2 y_3)^{1/3}\)
  • COmbined, G = \((x_1 y_1 \cdot x_2 y_2 \cdot x_3 y_3)^{1/3}\)
  • Thus \(G = G_1 \times G_2\)

GM with Log

\(GM=Antilog(\frac{\sum \log x_i}{n})\)

or

\(Antilog(\frac{\sum f_i \log x_i}{\sum f_i})\)

Mean of Variables

  • If \(y = a + bx, \bar y = a + b \bar x\)
  • If \(z_i=ax_i+by_i, \bar z=a \bar x + b \bar y\)

Example Problems

AM-GM

AM and GM of two positive numbers are 25 and 15, respectively. Find HM and the numbers. (click to see)
Solution (i)
Solution (ii)

Input Correction

The mean of 200 numbers was 50. Later it was revealed that two observations were incorrectly given as 92 and 8, instead of 192 and 88, respectively. Find the correct mean.

Solution

Sum of f(x-k)=0

If \(\sum f_i(x_i-k)=0\), what is the value of k?

Answer

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