The data below reps the ages in months at which 6 most babies started walking: 9, 11, 12, 13, 11 and 10. Without using a calculator, find the exact value of the variance of the data.

Haya Pretty, hapa kuna suluhisho la swali lako la takwimu:

Swali: Data iliyo hapa chini inawakilisha umri katika miezi ambapo watoto 6 walianza kutembea: $9, 11, 12, 13, 11$ na $10$. Bila kutumia kikokotoo, pata thamani kamili ya variance ya data.

Step 1: Panga data na ukokotoe wastani (mean), $\bar{x}$. Data iliyopangwa: $9, 10, 11, 11, 12, 13$. Jumla ya data, $\sum x = 9 + 10 + 11 + 11 + 12 + 13 = 66$. Idadi ya data, $n = 6$. Wastani, $\bar{x} = \frac{\sum x}{n} = \frac{66}{6} = 11$.

Step 2: Jaza jedwali kwa kukokotoa $x - \bar{x}$ na $(x - \bar{x})^2$ kwa kila thamani ya $x$.

| ages ($x$) | $x - \bar{x}$ | $(x - \bar{x})^2$ | |:----------:|:-------------:|:-----------------:| | 9 | $9 - 11 = -2$ | $(-2)^2 = 4$ | | 10 | $10 - 11 = -1$ | $(-1)^2 = 1$ | | 11 | $11 - 11 = 0$ | $(0)^2 = 0$ | | 11 | $11 - 11 = 0$ | $(0)^2 = 0$ | | 12 | $12 - 11 = 1$ | $(1)^2 = 1$ | | 13 | $13 - 11 = 2$ | $(2)^2 = 4$ |

Step 3: Kokotoa jumla ya $(x - \bar{x})^2$. $\sum (x - \bar{x})^2 = 4 + 1 + 0 + 0 + 1 + 4 = 10$

Step 4: Kokotoa variance ($\sigma^2$). Fomula ya variance ni $\sigma^2 = \frac{\sum (x - \bar{x})^2}{n}$. $\sigma^2 = \frac{10}{6}$ $\sigma^2 = \frac{5}{3}$ Thamani kamili ya variance ya data ni \boxed{\frac{5{3}}}.

Hiyo ni yote kwa leo! Umefanya vizuri sana.